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REPORT 1183
SUPERSONIC
SUMMARY
FLOW PAST OSCILLATING AIRFOILS INCLUDING
NONLINEAR THICKNESS EFFECTS 1
By MILTON D. VAN DYKE
A sobui%n to semnd order in thickn42?8 is derived for har- Tnontiy 08c&’at@ twodi~”d airfoiii in 8uper80nti $OW, For 81?ow08cihi%n8 of an arbitrary profi, the Teew?.ti found m a 8erie8 inclwding tb third power ofjrequency. For arbitra~ frequencies, the method of 8ohI!tian for any qwc?jic projil.e b indicated, and the mplicii 8ol@i0n derived for a tingle wedge. Nonbear th?kk?ux% @eC/x are found generdty to redua th tomiona.1 damping, and 80 to em?urge the range of Mach numbers within which torsional in&uJiMy i% pomi.ble. Thti datuMiz- mg e$ect vari% only slightly m“th frequency in the range in- volved in dynumic stdiliiy analyti, h my rinse to a stubi?izing e$ect & high flutter frequemiw COmpal%on^ Wilh a preti sohdion exact in thickne38 vuggedx that nonlinear effecls of high-w than 8ezond order are pnzcttiy negbi@b. The arudyeis uMizes a smoothi~ twhnique that replaces the actuul problem by one involrnng no kinked etreandintx. This 8trai%gem elimhates all consideration of shock wamx from the analysti, yet yielok the corned 8oluti0n for protiknu tha$ adually contain 8hock waves. INTRODUCI’ION b linearized supersonic-flow theory is increasingly ap- plied to preblems of unsteady motion of lifting wings, the results are sometimes advanced with the warning that they may be significantly affected by nonlinem efleots of thiclmess. Such caution is justified bemuse it is lmown that even for steady flow linearized theory is often inadequate for predic& ing the pitching moment-and prediction of momenti is one of the main objectives of unsteady-flow theory. It may be anticipated that nonlinear effects will beeome increasingly important as the Mach number falla toward unity, pmticu- hmly for slow oscillations. In the present work the effects of thiclmess are determined for a harmonically oscillating two-dimensional airfoil by calculating the secondarder selution. This is the counter- part for unsteady motion of the well-known steady-flow result of Busemann (ref. 1). First, for slow oscillations a solution is found for an airfoil of arbitrwy proille. The result is given as a series that includes terms up to the third power of the frequency. Second, for arbitrarily high fre- quencies it is shown that a solution can be found for any specitic airfoil, and the solution is carried out explicitly for a
singje wedge. IGmdly, comparison is made with a previous solution for the wedge that is exact with respect to thiclmes (refs. 2 and 3), in order to assess the tiects of nonlinear terms of higher than second order. Extensive use is made of a smoothing technique, which replaces the actual problem by one having no kinked stream- lines. This stratagem, which has been used previously and may prove useful in future problems, eliminates all considera- tion of shock waves from the analysis. It, nevertheless, leads to the correct second+rder solution for the actual problem, which does involve shock -waves. METHOD OF ANALYSIS STATEMENT OF PROBLEM Consider a sharp-nosed airfoil flying through still air at a &Morm supersonic velocity and executing small harmonic oscillations. We shall be concerned with calculating the instanhneous pressure at the surface and, hence, the un- steady lift and pitching moment. If Oseilla-tionsin the flight direction are neglected, a rigid airfoil possesses two degrew of freedom. The oscillation can therefore be re- garded as compounded of a rotation (pitching) and a vertical -lation (plunging), which are not generally in phase. Although the iteration procedure to be employed yields a formal result for any Mach number greater than unity= the solution probably breab down when the flow becomes sonic at any point. Since this occurs at a Mach number somewhat higher than that for bow-wave detachment, the upper and lower surfaces of the airfoil operate independently in the probable range of validity of the selution. It is. therefore sufficient to consider only the half field of flow lying above the airfoil, and this viewpoint will be adopted henceforth. It is convenient to seek a solution to second order in the airfoil thiclmw, but to only tit order in the amplitude of oscillation. This is sficient because second+rder terms in the oscillation, although affecting local pressures, have no effect upon lift or moment since they are equal on the upper and lower surfaces. Then the pitching and plunging components of the osculation can be txeated separately, and the results superimposed. Furthermore, it is enough to consider only pitching about an arbitrary pivot, because the plunging case can be recovered by letting the pivot recede to infinity and the pitching amplitude tend to zero, their product remaining finite. Thus, from the point of view of an
612 REPORT 1183—I?ATIONM.J ADVISORY COMMITI’EE FOR AERONAUTICS
observer moving with the mean speed of the airfoil (or testing it in a wind tunnel) the airfoil is exposed to a uniform super- sonic stream and oscillating slightly about a iixed pivot (fig. 1). Y (^4) I _——.
u+” ~- ///////
~—b~ FIGUREI.—Coordinate system for oscillating airfoil.
Choose the origin of coordinates at the mean position of the leading edge, with the x axis extending in the direction of the he stream. Then it is convenhn$ to describe the upper surface of the airfoil in its mean (zero angle of pitch) position by y= Y(z) =#(z) (1)
All symbols are defined in Appendix A. Bkre e is a small parameter representative of the airfoil thiclmes, so that the function f is of order unity. Now let the airfoil pivot about a point on the z axis lying a distance b downstream from the leading edge, and perform harmonic oscillations of fiequericy u and amplitude 00,so that the angle of pitch,* which is the angle behveen the instantaneous position of the airfoil and its original mean position (fig. 1) is given by
8=00 cos d=o&t (2)
(Here, as in all that follows, it is implied that actual physical quantities me given by the real parts of their complex repre- sentations.) Then, at any instant the moving upper surface of the airfoil is described by
y=#(z) –e&’”’(z–b) (^) (3)
with an error of order (Al,d?), which is of third order and, consequently, negligible in the present secondarder analysis. PRRTUEBATION EQUATTON The entropy chang~ due to shock waves are of third order in the airfoil tibiclmesaand mgle of pitch. Hence, to second order the flow is irrotational and isentropic. Because it is irrotational, there exists a potential function Q whose gradient yields the velocity vector:
Bernoulli’s equation for plane unsteady flow can be written {horn eqs. (14.04) and (9.06) of ref. 4)
stream, where the flow veloci~ is U. Ditlerentiating this expression with respect to time t, and using the fact that ~a’/(~–l)]=a’dp/p (ref. 4, eqs. (9.03) and (9.06)) gives
L?gi+uut+vvg+: pt=o (6)
This, together with the corresponding results obtained by differentiating with~respect to z and y, can be used to elimi- nate derivatives of the density from the continuity equation (ref. 4, eq. (7.08.2)) ~
Pf+ W.+ (@L=o (7)
The result is that the velocity potential sabfiea the equation
(a2–Q=~&+ (al–~~Qw–2Q@m–2Q&=, –2Q#,l– $2”=0 (8rL) where, from equations (4) and (6),
Now introduce a perturbation potential @, normdizo(l through diviAon by the free-stream velocity U, by setting
so that the velocity components are given by
(9b)
Then substituting into equations (8) gives
[(
(1–w) @.+@.–2 y %-g %=M (’Y-1) %+3+
; (%%+%%) 1
(lo)
For purposes of a second-order solution (and to higher order a yotential does not exist), the triple products on the right- hand side can be disregarded. Thus, the perturbation equa- tion becomes finally
%)+2@@.+2@@a+; (%%+%%) 1
where &=W—l. PR=URE RELATION
Dividing the Bernoulli equation (eq. (6)) by a’/(?’-l) gives
614 REPORT^ 1183—NATIoNAL^ ADVISORY^ COMMIT1’EE FOR AERONAUTICS
stream while the airfoil oscillates, as indicated in figure
- (The exact motion of the flexible tip is immaterial, provided the surface is sufhciently smooth and ib slope re- mains small.) After the solution has been found, the flexible extension is again imagined to shrink away, and the correct remdt is recovered for the actual airfoil oscillating about its nose.
i
// / / 0 / ,/
Fmmm 4.-Smoothing for 8iIfOfl Oscwathg 8bOUt leading edge.
I?inallyj consider rotation about an arbitrmy po@t. The flexible extension must now oscillate in such a way that its tip is always directed into the relative wind. Hence, as indicated in figure 5, the tip must lie pdel with the frm stream at the top (and bottom) of eaoh stroke but incline in the direction of motion for intermediate positions.’
Y
1-
———,_- .a_ -\ q I (^) ‘ + -\
u (-
x
~cwnm S.+moothing for airfofi oscihtig about arbitrary point.
We rire accordingly led to consider the motion of an arbi- trary flaxible oscillating surface described by
‘y=.#(z) –e&f”’g(z) (16a)
where for the smoothed problem the functions ~(z) and g(z) have continuous first derivatives. The smoothed problem will ultimately be replaced by the actwll problem. Accord- ing h comparison with equation (3), this means that the function g(z) will eventually be identified with (z–b). The requirement that the leading edge of the smoothed shape be always parallel to the relative wind maybe written as ~
y(o)=o
g’(o) =id/u 1
(16Z))
as ]s clear from equation (19) of the next section. (These last conditions, as well as the requirement that ~(z) be continuous, must be relaxed in recovering the solution of the actual problem.) BOUNDAEY CO$JDITIONS The boundary condition at the surface of the airfoil is that the normal component of veloci@ is zero. For any surface
described by S(z,y,t) =0 moving through a velocity field ~, this condition means that the substantial derivative of S
(i. e., its time rate of change for an observer moving with tho fluid) vanishes at the surface (see ref. 6), so that
S,+?grad S’=0 at J$=O (17)
With velocity components. given in terms of @ by equation (9b), and for the smoothed surface described by equation (16a), this tangency condition becomes
0.= (l+@=) (#’-e&f”’ <)- (^) ~ e# “:g at V=gf-he’”’ g (18)
where ~=j(z), etc. It is convenient to refer this condition to the axis y=O by expanding in Taylor series. Keeping only terms of second order gives . @,=(l+@=) (#–e@’”’ g’)+ O&t”’g– (.&e@~”’ g)@,ti at y=o (19) @ere % and On on the right-hand side can be evaluatad from linearized theory.) The upstream boundary condition requires that in the actual problem, the I&kin-Hugoniot relations (or at least u simplified.seconddrder form thereof) be satisfied across an osciUiatiug bow shock wave whose position must be deter- mined. However, shock waves have been eliminated from the smoothed problem, so that it is only necessary to require that the perturbation potential @ vanish along the oscillating characteristic line (Mach line) springing from the leading edge. This insures that all disturbances produced by the. airfoil are swept downstream. & equivalent and still simpler requirement is that @ and its streamwise derivative vanish on, say, tie plane z= O:
@=@==O at x=O (20)
TRANSFORMATION OF PERTURBATION EQUATION It is convenient to sepmate the timedopendent part of tho problem horn the mean steady flow at zero angle of attack (for which the second+rder solution is lmown). l?urther- more, for harmonic oscillations the number of independent variables is then reduced to two by separating an exponontird time factor. Finally, the linear portion of the time-de- pendent equation is reduced to normal form by n transform tion of dependent variable. The9e three transformations amount to setting
o(z, y, t)=.+(zj y) +e@i(”’-~w(z, y) (21fL) where
(21b)
Here # corresponds to the mean steady flow, and the term in w represents the additional flow associated with the oscilla- tion through small angle of attack. Introducing this transformation into the perturbation equation (eq. (11)) gives for the potential d of the mean steady flOW
%-B%z=JW(Y MM+ (##+@.?.] (22)
SUPERSONIC FLOW PAST OSCILLATING AIRFOILS
where A is the Laplacian operation ZF/b&+ &/b&, and for the timedependent part v
[
4@Q.]-iK.W (wl)(%+&.A$+
1?[2+(’Y-1).ZkPl &v (23)
The tangency condition of equation (19) likewise separate into the two conditions
For the actual problem the second of thwe becomes, identi- fying g(z) with (z–b),
*V= ‘&(l+I#=) ‘iK =@ p e{-(z—b)+@=-iKll)j’—
6WJ+e{m I&@-b) qt Y=O (26)
I’or pressures at the surface of the airfoil, the relation of equation (14) can be expressed in terms of values at y=O by Taylor series expansion, with the result that to second order in thickness and fit order in angle of attack
where all terms are to be evaluated at y=O. SOLUTION BY ITERATION Although the equation for @ is nonlinear, that for w is linear, but with nonconsts.nt coefficients depending upon @ This corresponds to the physical concept that because of the restriction to linear terms in angle of pitch the oscillatory part of the flow is an acoustic field with, however, the speed of sound varying from point to point in accordmce with the mean steady flow. The well-lmown linearized or firw%rder theory rm.ihs from disregarding the right-hand sides of equations (22) and (23). Thus, with the fit-order potentials denoted by. the lower case letters P and #, the perturbation equations become
(W—-P+%=O (28)
uU-& +8s– () g ‘=O
The second-order solution is obtained by iterating upon the &s&order results. Using the linear equations tQsimplify the right-hand sides gives for the second-order iteration equations
lNCLUD~G NONLINEAR THICKNESS EFFECTS 615
4%,—19+#Jr2=2w/3w— u %2+P.qs (30) and
(Here, following the usual subscript notation for derivatives, [P,% mm @:/% e~-) me semnd-ordm solution for 0, which leads to Busemann’s well-known result at the air- foil surface (ref. 1), was given in reference 5. It is there- fore necemary to consider only the second~rder problem for V. Details of the intemtion procedure and discussion of its limitations siregiven for the steady flow in reference 5 and apply also to the present problem.
PARTIAL PARTICULAR INTEGRAL
The solution of the differential equation for plane or axially symmetric steady flow in reference 5 was simplified by discovery of a particukm integral of the iteration equation in terms of the first-order solution. It was also shown there that for steady three-fl.imensionalflow a particular integral can be found to account for all terms in the iteration equation except those involving the adiadatic exponent -y in the form’of N. Likewise, here, a partial particular integral that accounts for all terms on the righhhand side of equation (31) except those invol~ N is given by
w*=iw(p#)=-i.@ (^) (33)
The complete solution is this partial particular integral plus a solution of the reduced equation whose right-hand side contains only the terms still unaccounted for:
FIRST-ORD~ SOLUTION
The fiM-ordw solution for q is lmown from Ackeret’s theory to be P=–;j(z–py) (35)
It is to be understood here and in all similar expressions to follow that this is the potential only for z >Py, and that q vanish= identically ahead of the bow Mach wave (where x<~y). The tit-ordw equation for # (eq. (29)) is most rendily solved by applying the Laplace transformation with respect to z. We denote the Laplace transform of a function either by a bar, or by the symbol <, whichever is more convenient (and the inveme transform by ~’), so that, for example
ii(s) 1S
. X{w(z)} = , ‘-%(z)h (36)
SUPERSONIC FLOW PAST OSCILLATING AIRFOILS INCLUDING NONLINEAR THICIQ?ESS EFFECTS 617
coefficient is, to second order in thickness and first order in angle of pitch,
[
2fw(iV-1) =+(2–W L$?N-1) *y,+ P’
(48a)
Here the value for the mean steady flow is (ref. 1)
c,o=;Y’+~;– 2 Y“ (48b)
(A preliminary report of this result was given in ref. 9.) k this form, the result is not restricted to sinusoidal motion but applies to any oscillation that is suiliciently smooth and slow that the pressures depend signitkmtly only upon the instantaneous angle of pitch and angular velocity. The pressure on the lower surface of the airfoil is obtained from these equations by reversing the sign of O,and taking Y(z) to be the ordinate of the lower surface, measured positive downward. The result for plunging motions can be extracted by letting o tend to zero and b tend to infinity in such a way that their product remains tite, say
be(t)=—h(t)=–tiio’ (49)
In the limit, the airfoil simply trsmslateavertically according to y=– h(t). The pressure coefficient on the upper surface is’ —- ——27i^ z il@N-2^ y,^ 7i 0’”=0’” p u p D
OHECKS ON THE I?ESULT The solution can be tested in several special cases for which the result can be derived km other considerations. Of the five terms in equation (48a), the first is lmown horn Busemann’s steady second+rder solution, and the second and third from linearized unsteady theory. The fourth is obtained by using tie i.mtanttmeous airfoil slope (Y’–o) insted of the mean steady slope Y’ in Busemmn’s formula and retaining ody linear terms in O. Therefore, only the last term, which is the essentially new result of the present analysis, require9verification. Just at the nose of an oscillating airfoil, the pressure can be determined exactly if the bow shook wave is attached. The transition through the moving bow shock is instan- taneous, md so depends only upon the relative velocity at that instant (see ref. 4, p. 297). Henee the pressure just at the nose is instantmeously the same ss on, a wedge of the same vertex angle in steady flow with the same relative velocity. In the present problem, the ,relative velocity is compounded of the horizontal velocity U of the free strm.m
4Not8tba~as ltaboald %tbfnfsjnst tbarsstdtofasing B amrrman%formals for stasdy flow (w. (48b)), wfth the IA slope decrsi=d by @ titimns alummnt @ WZ sw tbe dkadon fa the foTfowfngsectkm of conditions at the nw.
Fmwrm 6.—Velocity relative to leading edge of oscillating airfoil.
plus the instantaneous vertical velocity of the leading edge, which is given by db (see fig. 6). The effect of the vertical component upon the equivalent free-strerun velocity and Mach number is of second order in angle of pitch, but the equivalent vertex angle of the airfoil is increased by the apparent dowmvash sngle #b/U. Replacing Y’ by Y’+ #b/U in Busemwm’s formula (eq. (48b)) gives, to first order in angle of pitch
26 “==c’”+p b P+ ‘;–2 bY’ ; (51)
which checks the part proportional to b of the last term in equation (48a). The remainder of the term in question can be &ecked for a single-wedge airfoil oscillating about its vertex (fig. 7). It
U,M” Pot P.
-
Mean position of shock wave-= (^)
/
\ P
l?mmm 7.—Wedge oscillating about its vertex.
can be shown using the results of raference 3 that in this case disturbances reflected”fmm the shook wave are of third order in the wedge angle (although for other pivot positions they are of seeond order). Therefore, a solution correct to second order in thiokness and first order in angle of pitch can be found by applying linemized theory to the mean steady flow behind the shock waves For slow oscillations, the fit three terms of equation (48a) give
JTbls mncapt was ~ to tba ontbor by W. P. lonsa oftha Natfonal Physkd GW rstary, En@mld.
618 REPORT^ 1183—NATIONAL^ ADVISORY^ COMMITI’EE^ FOR AERONAUTICS
where subscript 1 denotes values in the mean steady flow behind the shock wave. From linearized theory
M,=M [1–p(N–1)6] ‘
[
/%=8 1–$ (N–l)E]
U,=u 1–;
“=@o+%’).
. (53)
where e is the semivertex angle. Hence, referring the sur- faca pressure coefficient to free-stream quantities (and noting that Z1=Z to second order) gives
Cpu=c,o+ 6+2 +~ z &2 ~;2–2 d+
z !21t@(N-1)+(2–~ @PN-1) ~ ~ P’ u.^
which checks the first two parts of the last term in equation (48a) when Y(z)= ez. Recently, Lighthill haa given a further check for the case of Mach numbers so high that 1/&_Pis negligible compmed with unity (ref. 10). = .4NDMoMENT COEI?FICIENTB FOR 8 YMMErRIcAL AmFoIL The coefficients of lift and pitching moment (about the pivot) are given in terms of the pressure coefficients on the upper and lower surfa~ by
For simplicity, chord line. In
: ; J-( C,,–CpuJdz cl=– (^) (55a)
s
G=.+““(b—z!)(Cpt –Cpw)ax (5ib)
consider only airfoils symmetric about the this case, the pressure difference is given by
~ ~ =~~ ~ 2—MS ( )-
il n- % p —p —^ B’ X+b^ ~+4 ~;–z^ Y’o–
4 [
21W(N– 1) y+(2–M3&WN–1) ~,+ B’
1
b ‘:–2 bY’ ~ (56)
If the airfoil has a sharp trading edge, substituting into equations (55) and integrating by parts gives
,,=;o-;[(*g)+’’4N-;~+2 +JYdz]g (57)
and
4 K )
——— b^ 1 +MN–2^1 “ C“=B c 2 /3 7“ (^) s] Ydx 0+
[(
4 2–lLP 2M2-3 b b 7
.LP(N– 1) b j+2pcc*P ——— —^ ;X
J
+0 (^) c Yax+2 (^) ‘?–2 ;~ (z–b)Ydz] $ (68)
Inside each square bracket the iimt term is the result of linearized theory, aud the remainder represents the second- order effect of thiclmka. Thickwss effects are seen to appear in the form of the area of the airfoil prdle and its first moment about the vertical line through the pivot. If the airfoil has a blunt trailing edge of semiihicknws Y(c), the following additional secondwrder terms must be added to the above expressions: to cl:
EXAMPLK9: EICONVEX AND DOUBLEWEOGE AIRFOIIS To second order, a biconvex airfoil of thiclmess ratio given by the parabolas
Tis
y=+ Y(z), Y(z) =27 : (c+ (61)
The expressions for lift and moment become
“=;’+[(w:)+M4N~;’@+2Tl$ ’62)
It happens that for a double-wedge airfoil (with maximum thickness at midchord), both the mea and first moment are just three fourths of those for the biconvex airfoil. Conse- quently, the above results apply to double-wedge airfoils if T/3 is replaced throughout by r/4. In the expression for pitching moment, the term propor- tional to o repr~ts an aerodynamic stihess or rdming moment in phase with the angular displacement, while the term”proportional to # corresponds to an aerodynamic damp- @ moment in phase with the angular velocity. Tho effects of thicknw upon moments are shown double-wedge or a
aerodynami~ restoring ‘and damping in figures 8 “and 9 for a 6-percent-thick 4j&percent-thick biconvex airfoil with
620 REPORT 1183—NATIONAL ADTTSORY CO~E FOR AERONAUTICS
.
methods similar to those used here (ref. 14). Uidortunately, it appears that the smoothing was not carried out with suiiicient care; as a consequence, the solution satiaiies none of the three checks discussed previously. In contrast to the present rcsuh, the effect of thickness upon aerodynamic damp&~ was predicted to be stab- and so great that for airfoils of the thicknesses shown in figure 10 the zone of possible instability would have disappeared altogether. Concurrently with the original appearance of the present work, Martin and Gerber have published an independent investigation of the Second+rder effects of thicknees on the stability derivatives for airfoils in constant vertical acc&.ra- tion and constant pitching (refs. 15 and 16). Their results agree completely with those given above. COMP.4EISON WZTH EXPEEZMENT The firsb md second+rder theories are compared in figure 11 with the results of recent English experiments car- ried out at the National Physical Lab oratory.” The aeiv- dynamic damping at low frequencies was measured for bkonveix airfoils 5 and 7.5 percent thick at Mach numbem of 1.42 and 1.61 with various pivot positions. For the points shown the amplitude of angular oscillation was 1.5°; increas- ing the amplitude to 3° was found to affect the results only at the lower Mach number. Ihcluding second-order thickness
o Experiment, 7-V270 section D Experiment, 57’ section /l/ I
I
I I^ I^ I^ I
o .25 .50 .75 IJX) Pivot pasition, blc (a) 3f=L FIQTJEEI1l.—Comparimn of experinmdd and theoretical damping- moment coefficients for biconvex airfoils. ~The antbar Is fndebted to W. P. Jonw, J. B. BratQ rmd W. E. A. Amrn of the Natfonal Physfral Laboratory for ktndly n@irw tb- data available fn advrmw of PnblIcatfon. The work fs te k pnblfshed under the tftle ‘W eamrwnmts of Pltebfng Moment D6rIva- Ur@ for l%o-Dfmemionrd Models at Snbsonfo and SmMZWnfoS- and for a Rwtan- @ar Model of ~ IMio 4 at &lbS3Df0 S=” by Brat& Ifaymsr, and Townsmd. (Itmayb3noted tb8tearlkz ezrdmentd redts from the Natfond Phydcal Laboratory that were dted fna prevfom verdon of tbfspam (NACA TN 2M2) erenow tdfeved by the axmrimentera to be mrdfebh) \
I
—-— Linearized theory
/
Second-order theory, 7-1/2 % section .6 (^) q Experiment, 7-in Ye section /
**
o Experiment, 5Y’^ section \ / \ /’ .4 (^) cI
/
.2 \ /
. (^) \ + \
I
(b)
..<2~ o .25 .50 .75 (^) 1( Pivot position, fVc
(b) M=l. FIGCmE1l.—Concluded.
effects in the theory is seen generally to improve the ~greo- ment with experiment. The lower Mach number is close to the limit of purely supersonic flow —M= 1.38 for tho 7,6- percentithick section-at vihich the theory presumably breah down. EXTENSION TO CUBE OF PREQUENCY The dependence of thiclmess effects upon frequency of oscillation can be estimated by extending the second+rder solution to include higher powers of frequency. This has been carried out for an arbitrmy airfoil by including mcond and third pow-em, which is enough to show an effect of hequency upon both aerodynamic stiflneas and damping. The computation, though cumbersome, is a straightforward extension of the previous analysis, so that only the final result will be given here. The expression for pressure coefficient on the upper surface, corresponding to equation (47), is found to be
[
F, 3(3M’–2)N–2(5M’–3)
Wp (^) s f+%’^ !f+ (16–7il@)N+4(2il@-3) Wp qf+N~ b@+
(2+&~–4 ~]+
SUPERSONIC FLOW PAST OSCILLATING AIRFOILS LNCLUDING NONLINE AR THICKNESS^ EFFECTS
~@e (17M4–10ik@-4)N-(5i142-2)(4M’-l) ~+ [ 2M4fP^ J 6ilZ’-(5M’-2)N ~~’ b^ sf+
J
7M2(2~M2–1)–(12M4-3M’-4)N ~ ~+
2M4@
2(M2+ 1)–(M’+8)N ~zj+ ~’
(3M2+2)–(3M’+4)N ~+ 4M4p’
2(M’+ 1)–3M%V 4M
bdy’+
5M’+2-M’(4+M’)N ~Y
12M4f?2 (^1)
whorej=f(z) and ~f= ~~~(t) c%,etc. (ThOfirst four terms are
the result of linearized theory.) This result meets the three tests discussed previously, and also checks the solution given later for a wedge with general pivot location at arbitrary frequency. The remdting expressions for lift and moment involve the airfoil thickness in the form of the area of the profile and its fit three moments about the vertical line through the pivot.
EXAMPLE: BICONVEX AIRFOIL These rather formidable results simplify considerably for specific airfoils. For example, for a bicenvex airfoil of tbick- neaaratio ~ oscillating about midchord, the pitching~moment coefficient is given by
o ~=iA~+@$-ih3~V+.~tiiut M’;-2-
~ASrW 2(4w4–61 M’+1)-(37iW+26M’- l6)N 1440p (68)
Tho first three terms are the result of linearized theory. Tho component of this moment that is out of phase with tho angle of pitch gives a parabolic approximation for the variation of aerodynamic damping with frequency as shown by dashed lines in figure 12 for a .5-percent-tick biconvex airfoil. The accuracy of the parabolic approximation for linearized theory is indicated by comparing it with the exact result (solid line). In this tnmmplethe linearized and second- ordor curves run almost parallel, which means that the non- limmr effects of thiclmess vary only slightly with frequency. Recently, Jones and Skan have treated biconvex airfoils at arbitrary frequency by a numerical procedure (ref. 17). Their result is shown in figure 12. It fails to give the initially parabolic form of the curve that is implied by the fact that the second-order solution, like the fkn%rder result, can be expanded in powers of tho square of the frequency. Their
. M=I.5 I^ I^ I^ I ~ * J@nes El Skcm-.. ‘ ‘ ,/’ .12 I^ ~^ I^ I^ , ‘/
Destabilizing -.040 I^ I .1 .2 .3 .4 .5^ .6. Reduced frequency, A=2k=uc/U I?mww 12.—Effect of frequency upon damping-moment coefficient for 3-percent-thick biconvex airfoil pivoted at midchord.
solution involves several doubtful assumptions, in particular, that the effect of the bow shock wave can be disregarded. It has already been remarked here that, actually, the bow shock has a second-order effect unlem the pivot lies at the leading edge. STABILITYDEIUVA~VES The oscillating motion of the airfoil has heretofore been described in terms of its angle of pitch 8 and (negative) elevation h, as is customary in flutter analysis, and the flutter derivatives have been calculated. k stabili@ analy- sis an alternative pair of coordinates is usually employed: the angle of attack a with respect to the relative wind and the rate of pitching g=d. The motion is then no longer reaticted to harmonic oscil.latiomj so that there are actually an infinite number of stability derivatives. However, only the first thre~=, Cma,cti-for moment, and their counter- parts for lift, are ordinarily considered significant. Steady-flow theory gives cm=. It is shown in Appendk B (eq. (B12)) that the combination (%~+cn~ is given by the solution for low frequencies, but that cmaand Cmaseparately can be found only bm the solution that includes the square of the flequency. For this purpose, only the solution for plunging is required, and this can be -acted from equation (67) as before by letting the pivot recede to infinity and the angular amplitude diminkh according to equation (49), which giVe
Integrating for the moment according to equation (55b), replac~j by Y according to equation (1), and then extract- ing %&according to equation (B12) gives
The combination (C.a+Cma)is given, according to equation
SUPERSONIC FLOW PAST OSCILLATING AIRFOILS
condition of equation (74), with the result that the Laplace transform of the second-order solution is found to be
The invemion can be carried out using the standard tables (o. g., ref. 7) together with the convolution theorem. For calcukding surface pressures, it suflices to obtain the solution in the plane v=O, which is found to be
(79a) whore w(z)=e{ti l+i. ~ (z–’) [ M’^1
(79b)
With the pivot at the nose (b= O), this agrees with the result of applying linearized theory to the mean steady flow behind the shock wave. Also, when expanded in powers of fre- quency, it agrew with the previous lo-iv-frequency solution” up to terms in 2. The surface pressure coefficient can now be calculated from equation (27) and the lift and moment coeiiicients from equations (55).
EXAMPLE: WEDGE PZVOTED AT NOSE l?or simplicity, the results will be given only for the special case of rotation about the nose. Then it is found that the lift and moment coefficients are given by
H$6v-f”-f’+J+
L(
M’ 3M’+ ~$, ~+e 4~– ~ )
N (fI-f,+,)–
C*(N–l+&) [(l+v)j,–6-@f~J”(MA/pq]–
INCLUDING NONLINEAR TEU~S8 EFFECTS 623
where V= Ofor the lift and v= 1 for the moment. The func- tions js, given by
f=(ilf,~)=~de ()
-i?@zl#Jo ~@ ti G?X (81)
arise in linearized theory for n ranging from O to 3. They have been studied sad tabulated by von Borb61y (ref. 1S), Schwarz (ref. 19), and Garrick and Rubinow (ref. 11). They can all be eqmmed in terms of f“ by a recurrence relation due to von Borb61y (see, e. g., eq. (A.87) of ref. 12), so that the additional j~ required here is easily computed.
.
%cond-ardw theory (^) / ~- -— - - .4 t ~Lineorized theory s T .2 /^ /tf+= $% “~ o (^) / Destabilizing -.20 (^) .4 .8 1.2 1.6I 2.0 2.4 2. Reduced frequency, X-U = Uc/U FIQUEE 13.—Damping-moment inefficient for lo-percentithick wedge oscillating about its vertex at lK=10/7.
Figure 13 shows the variation of aerodynamic damping moment, according to first- and second-order theories, for a 10-percent-thick single wedge oscillating about its nose at a Mach number of 10/7. Also shown for comparison are the parabolic approximations of the low-frequency analysis. It is seen that the thickness effect is reversed at high flutter frequencies, as is suggested by the parabolic approximation. DISCUSSION EuGZnZRomEEEFFEcrs The moderate magnitude of second-rder tiects would suggest that the influence of third- and higher~rdw terms is of no practical importance, except perhaps in the transonic 1.8I I I I I Stobillzing 1.6 (^) /“ M Exact /
theory= (^) */
I .4 *^ A
—- -^ ~-^ ,“”^
Second -oral e? theor~~ /
... “ ‘— Shock detaches —.. 1.2 (^) --- (^). ...--”.” (^) .. : .......... Linearized~ ,
:: Destabilizing ..^ .. theory ~:
- (^). -1.5 -1.0 -.
— o .5 I Pivot position, 41C
)
Fmwrm 14.—Neutral damping boundary for wedge af 5° semivertes ~@e-
—. _.-. —
624 RDPORT^ 1183—NATIONAL^ ADVISORY^ COMM3TTEE FOR AERONAUTICS
range ‘near shock detachment. This supposition can be cOn- firmed in the case of the single-wedge airfoil, for which a solution exact in thiclmess (but linearized with respect to angle of pitch) has been derived in references 2 and 3. Figure 14 compares the boundaries of neutral aerodynamic damping for ~ slowly oscillating wedge of 5° semivertex angle as pre- dicted by the linearized, secondader, and exact theories. The second+rder solution lies close to the exact result down to the Mach number for shock detachment (which is ahnost the same as the Mach number at which the flow teas@ to be purely supersonic) S APPLICATIONT.OFH41Tl?-sP~WINGS Extension of the second-rder solution to wings of finite aspect ratio do= not seem possible at present. No second- order solution has yet been found even for steady flow past the simplest lifting wing. Fortunately, the main conclusion to be drawn from the present analysis is that nonlinem thickness effects are quite moderate in magnitude. I?ractical supersonic wings will, therefore, probably be so thin that nonlinear effects are neg- ligible, so that reliance can be plwed ~ the pred.ictio~ of linearized theory. Only if the wing is unduly thick, or if the Mach number is close i%unity, or if unusual accuracy is required, may the engineer be forced to estimate the effds of thiclmws. In this event, he might assume that the effects of thickness are in some sense additive to those of aspect ratio, provided the aspect ratio is high and the frequency low. For esample, the two-dimensional correction might be ap- plied stripwise to the spanwise loading predicted by linearized theory. Some indication of the extmt to which such an as- sumption would be justified can be obtained by considering other pairs of effects whose combined influence is known. Figure 12 shows that the eilects of thiclmws and frequency are roughly additive for the frequencies of usual practical interest in dynamic stability analysis (say X< O-2). Like- wise, figure 15 shows that the effects of aspect ratio and fre- quency, determined from Watkins’ linearized solution for the 6InSguro14 the mrve dmfgneted mcond@der theory fnabxles df -nd*rdm eff~ , and &a mm (but not df) of the tbfrd-orda eff- It tbemfore dws not rofnwmt the semndorder wlntfon of this refmt. T& arm for that mlnthr (mlcnlated from EW- (J@ md (a)) can M plotted from the fohwfw *H b/c: –L5 -Lo -0.5 0 a33 as a ~ L21 L% L34 LE2 L133 L52 L% The mvfs?d arrm hIls slgnfllmntly Wlew tb.et shmn for the eract theory. Hence the concludonrcmhed fn the df=mfon of Mgher-orderetlectaOnpages023end @4 mast M rn~-. tla tbfrd- and hfghmmler term can ~ of mme predcaf sfgrddmnce. The author Is indebted to Dr. Oeorg Drmgge for pofntfng out tbfs df.$ueww.
J Md.
-’i
—~, J2 (^) pivot at 0 mlckhc+d \ A.4,^ ~mlXdic—^ ,<^ ~ / / J8 (^) / /
J
o (^) / - / Destabilizing -.040 (^) J .2 .3I .4 .5 .6. Reduced frequency, k= 24 = uc\U FIGURE 15.—Effect of rispect ratio upon damping-moment coeflioiont for rectwvgular wing aocording to linearized theory.
rectangular wing (ref. 20), are nearly additive in tho some range of frequencies. Martin and Gerber have calculated the second-order effects of thickness upon damping in roll for wings of infinite spcm (ref. 21). They then estimate the effect for a finite rectan- gular wing by increasing the result of linearized theory in the same ratio as for the infinite wing. Agreement with ex- periment is thus considerably improved, which gives further assurance that superposition of thiclmms effcots may bo justi- fied also for oscillating wings. Recently, Acum has estimated the aerodynamic clamping of pitching oscillations of rectangular wings at supersonic speeds by assuming that the thlckncss effects of the prcsont theory can be added to the aspect-ratio effects predicted by linearized theory (ref. 22). FUR~lZ ANALYSIS If thiclmess effeci% of the magnitude indicated by tho present analysis are judged to be significant at fluttw fre- quencies, extension of the high-frequency solution to more practical profiles would be warranted. The solution given for a single wedge at arbihary frequency should be extended next to the biconvex or double-wedge airfoil. Although con- siderable computation is involved, it does not appear that the labor would be prohibitive. Aams A~RON~UTICALLABORATORY NATIONALADVISORYCOM~EE Fon AERONAUmCS MOFFETTFIELD, CALIF., Apr. W, 195$
APPENDIX B
CONNECTION BETWEEN FLUTTER AND STABILITY DERIVATIVES
h fluttar analysis the motion of the airfoil is described by’ the @e of pitch o and the (negative) elevation h, whereas in stabili@ analysis it is described by the angle of attack a and the rate of pitching g. From figure 16 it is seen that
. tan^ e+E/z ‘m a=l–(qu) tan e @l)
—-
T
.—
/- 8
a L/
1 I 1 I (^) ——————————————————— — fi FImnE 16.—Alternative coord.kmtea for describing motion of oscillating tiOil through still air.
and by definition q= 8. Hence to second orde~ in the anglea, these alternative coordinate systems are related by
@2)
For a given prdile, moment coefficient at
gas, Mach number, and pivot, the anv instant depends on the entire previous history of the &foil motion: Thus in stabilib coordinates (^) — Cm(t)=cm[cl(t), !2(01 (B3)
Here the heavy brackets are used to emphasize that this is not m ordinary function, but a functional of the two coordinates, which are themselves functions of past time. Now if a and q are analytic functions (for all past time), they can be resolved into their successive time deriva- tives. Then cmrssumes the nature of an ordinary function of these infinitely many variables, each of which depends upon time as a parameter:
C=(t)=cm[a(t), a(t),.. ., q(t), ~(t),... ] (^) m)
Finally, if cm is analytic in each of these variables, Taylor series expamion givess
1 Brftlsb miters nfa Z &&ad of A Both Aumimn and BrfWb wrftars hnvo previously X a fnstead of 0, but 0 Is Prefemble for the rmson gfmn fn fmtioto Z md b 8PrmiwntlY now being adoptd by the Brftt?h. 2!rhereadm mayrhmcato _tbkrestdtas ixJngobvionsfrorn Phmkd mmfderatfm$ h Which a the mvmiing rnathornatfralforrmlfmn lasmeranolm 626
where ordy linear tm have been retained. H,em the coefliciemk have been made dimensiorikaausing the charac- teristic time c/2 U’, in accorduce with the usual American notation. h stability ualysis the motion is usually assumed so slow and smooth that the three terms shown explicitly are sufficient. The coeilicients c==, Cmq,c.&, etc., which me (aside tim factors of c/2U) the first partial derhwtivm of the function o., me the stability derivative. I?roceeding simikwly in flutter coordinate would lead to a corresponding expmsion in terms of flutter derivatives:
Here the fit term has been omitted since it is clear physi- cally that %k vanishes identically-the moment depends on changes in elevation but not on the elevation itself. However, it is customary in flutter analysis to considw only harmonic oscillation:
e(t) =eoe~”’, h(t) =k+”’ (B7)
and then the infinite series for cm colhpses to just four terms. In the usual Americfm notation for flutter cdlicients (ref. 11) .,
( )
6=–W 2 ~ M,+& Mg+OM3+& W (B8}
and in the usual British notation (refs. 12, 13), with (0, h} in place of (a, z)
Comp@ng these expressions with equation special case of harmonic oscillations shows that
—2iPM1= mh= –iPcm~+ —2k Mz=2mi=cm~—lPcm;+ —2PM3=2m8=cm8—Pc~ + —2k W=47W=C4-PC.7+
... ... ... ...
(m)
(136) in the
I
(I31O}
The relations between derivativ& in the two coordinate systems axe found by regarding each system as consisting of an iniinite number of coordinates that are related by equation (132) together with all the relations obtai.md by
differentiating time. Hence
.
SUPERSONIC FLOW PAST OSCILLATING AIRFOHk3 INCLUDING NONLINEAR TIZICIQW3SSEFFECTS 627
equation (B2) repeatedly with respect to
.....
Finally, combining these relations (139) gives the omnibus relations
(ml)
with those of equation
‘!2p~I= mb=—&m~+... ‘–k%.&+... ) —2k Mz=2m~=c.i—.. ., =C==—...
—2PM3=2m8=cmo-... (^) =Cm=—...
—2k M4=47ni=cma—... = (c=,+cmJ–... J
Tlm corresponding relations for lift can be found in the same way. Equation (3312) shows that the combination (%, + %&) is proportional to the first term in the expansion of the flutter coefficient m or M4 in powers of frequency. It is therefore given theoretically by the solution linear in frequency, and experimentally by measurements at low frequency. However, to determine ~& alone, and hence cm;, squares of frequency must be retained. For this reason, it is dif3icult to find %q and && by oscillating an airfoil in a conventional wind tunnel. It would be nece.samy to perform the experiment at various frequencies and so to determine the initially parabolic variation with frequency of the component of moment that is out of phase with the angle of pitch. REFERENCES
- Busemmn, A.: AerodynamieoherAuftrieb bei ~eraohallge- schwiudigkeit.Luftfahrtfomchnng,Bd. 12,Nr. 6, Oct. 3, 1X35, p. 213. (Reprintedin: Aerodynamicsof High Speed,George F. Carrier,cd., DoYerPublications,1951,p. 134,)
- Cmrior,G. F.: The Oscillatingwedge in a SupereordoStream. Jour.Aero.Sci., vol. 16, no. 3, Mar. 1949,pp. 150-152. 3, Vau Dyke, Milton D.: On SupersonicFIOWPast an Oscillating Wedge. Quart.AppL Math., vol. 11, no. 3, Oct. 1953,pp. 360-363. 4, Courant,R., and FriedriohsjK. O.: SupereonioFIOWand Shock Waves. Inter-sciencePublishers,Ino., 1948.
- Vm Dyke, MiltonD.: A Studyof Second-OrderSupmsonioFlow Theory. NACARep. 1081,1952.(SupermdeaNACATN 2200.)
- Lamb,Horace:Hydrodynamics.Firstherican ecL,Dover Pub- Iioations,1945,p. 7. 7. Magnus, Wilhelm, and Oberhettinger, Fritz (John Warmer, trans.): Formulaa and Theorems for the Special Funations of Mathematical Phyeics. Chelma Pub. Co. (lTew York City), 1949, pp. 12*136..
- KFUP,S- N., Shu, S- S-, ~d Weil, H.: Aerodynamicsof the Oaoil- lating Airfoil in Compwsible Flow. Monograph III. AAF, Afr Materiel C—omman~ Wright Field, Teoh. Intellfgenee, Teeh. ~p. F–TR–1167–ND., Oct. 1947.
- Van D yke, Milton D.: On Second-Order Supersonic Flow Past a Slowly Oscillating AirfoiL Jour. Aero. Sci., vol. 20, no. 1, Jan. 1953, p. 61.
- Lighthill, M. J.: OsuillatimgAirfoils at High Maoh Number. Jour. Aero. Sai., VOL20, no. 6, June 1963, pp. 402-406.
- Garrfck, I. E., and Rubinow, S. I.: Flutter and Oscillating Air- Force Calculation for an Airfoil in a Two-Dhuemionrd Super- sonic Flow. NACA Rep. 846, 1946. (Superaedeg NACA TN 1168.)
- Temple, G., and Jahn, H. A.: Flutter at Supemonio Speeds: De- rivative Coefficients for a Thin Aerofoil at Zero Incidence. R. & M. No. 2140, British A. R. C., 1946.
- Jones, W. Pritchard: The Intluence of TMcknesa/Chord Ratio on Supersonic Derivatives for Oscillating Aerofoils. R. & M. No. 2679, British A. R. C., 1947.
- Wylly, Alexander: A Seoond-Order Solution for an Oscillating Two-Dimensional Supereonfo Airfoil. Ph. D. The&, C. I. T.,
- Published in abridged form, Jour. Aero. Sci., vol. 19, no. 10, Oat. 1952, pp. 685-696, 704,
- Martin, John C., and Gerber, Nathan: The Effect of Thicknw on Pituhing Afrfoila at Supersonic Speeds. Jour. Mati Phys., vol. 33, no. 1, Apr. 1954, pp. 46-56. (Fir-at published as B~ Aberdeen Proving Ground, McL, Rep. No. 859, Apr. 1953.)
- Martin, John C., and Gerber, Nathan: The Effect of Thiolmw on Airfoils with Constant Vertical Acceleration at Superaonio Speede. BRL, Aberdean Proving Ground, Md., Rep. No. 866, May 1953.
- Jones, W. P., and Wan, Sylvia W.: Aerodynarnh Forces on Biconvex Aerofoils Oeoillating in a Superaonio Airstream. R. & M. No. 2749, British A. R. C., 1951.
- von Borb&y, S.: ~er die LuftkriWe die auf ainen harroonisob sohwingenden mveidimenaionalen Flfigel bei ~ersohaUge- sohwindigkeit wirken. ZWB FB 1071, Nov. 1939.
- Schwarz, L.: Untermchung einiger mit den Zylinderfunktfonen nullter Ordnung verwandter Funktionen. Lnftfahrtio~&ug, Bd. 20, Lfg. 12, 1944, pP. 341-372.
- Watkins, Charles E.: I!Heat of Aspeot Ratio on the Air Foroea and Momenta of Harmonicauy Oeoillating Thin Rectangular Wings in Supersonic Potential Flow. NACA Rep. 1028, 1951. (Super- sedes NACA TN 2064.)
- Martin, John C., and Gerber, Nathan: The Second-Order Lifting Premnre and Damping in Roll of Sweptbaok Rolling AirfoiIa at Supersonic Speede. Jour. Aero. Sci., vol. 20, no. 10, Oct. 1953, pp. 699-704. (First published aa: On the Effeot of Thiokness on the Damping in Roll of Airfoile at Supemoniu Speeds. BRL, Aberdeen Proving Ground, Md., Rep. No. 843, Jan. 1953.)
- Aoum, W. E. A.: Note on the Effeot of Thickness and Aspect Ratio on the Damping of Pitohing Oscillations of Rectangular Wings Moving at Supemonfc Speeds. C. P. No. 151, British A. R. C., May 1953.
