Document
NASA Technical Memorandum 86320
Effects of Airfoil Shape,
Thickness, Camber, and
Angle of Attack on Calculated
Transonic Unsteady Airloads
John T . Batina
Lcrngley Research C m t e r Hamptorz, Virginia
NASA
Scientifir and Technical Inforn.ation Eranch
Contents
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Transonic Code XTRAN2L and Pulse-Transient Technique . . . . . . . . . . . . . . 2
General Airfoil Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Linear Unsteady Airloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Results for Three Airfoil .Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Mean Angle of Attack of 0" With No Camber . . . . . . . . . . . . . . . . . . . 4
Mean Angle of Attack of 1.0' With No Camber . . . . . . . . . . . . . . . . . . 4
Mean Angle of Attack of O0 With Camber . . . . . . . . . . . . . . . . . . . . . 5
Results for Three Airfoil Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . 6
Mean Angle of Attack of 0" With No Camber . . . . . . . . . . . . . . . . . . . 6
Scaled Mean Angle of Attack of 1.0" With No Camber . . . . . . . . . . . . . . . 7
Mean Angle of Attack of 0" With Scaled Camber . . . . . . . . . . . 7
Comparison ol" Effects Due to Angle of Attack With Effects Due to Camber . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
iii foil, NACA 64A010, and a supercritical airfoil, MBB-
Summary
A3, were considered. Both airfoils are AGARD stan- The effects of airfoil shape, thickness, camber? and dard configurations (Bland 1979). Agreement was mean angle of attack on transonic unsteady airloads found between the harmonic aerodynamic forces of the were investigated as calculated by the finite-difference NACA 64A010 airfoil at a, = 1.0' and the MBB-A3 air- computer code XTRANZL. This code provides a t i m e foil at a, = -0.5'. The agreement in forces occurred marching solution to the nonlinear small-disturbance with a Mach number shift of 0.01 and held true for equation for transonic flow. The harmonic airloads for the entire Mach number range considered. This obser- airfoil plunge and pitch motions were determined by us- vation suggests that similarities exist in tran~onic un- ing the pulse-transient method available in XTRANSL.
steady forces for airfoils of different shape or thickness Shape effects were investigated by examining the pres- at different Mach numbers or angles of attack.
sure distributions, shock locations, and unsteady cir- The purpose of this research study is to system- loads for three 10-percent-thick airfoils: NACA 0010, atically examine in detail the effects of airfoil shape, NACA 64A010, and parabolic arc. Thickness effects thickness, camber, and mean angle of attack on tran- were determined by studying a single airfoil shape with sonic unsteady airloads. Transonic aerodynamic calcu- three different thicknesses: NACA 0008, NACA 0010, lations were performed using XTRAN2L. Shape effects and NACA 0012. Angle-of-attack and camber effects were investigated by examining the pressure distribu- were studied by including mean angle of attack or by tions, shock locations, and unsteady airloads for three adding a simple parabolic camber distribution to the 10-percent-thick airfoils: NACA 0010, NACA 64A010, originally symmetric airfoils. Comparisons of unateady and parabolic arc. The Mach numbers considered alrloads for different airfoil configurations show simi- were M = 0.76, 0.78, and 0.80. Thickness effects lar results caused by variations in airfoil shape, thick- were investigated by considering one airfoil with three ness, camber, or mean angle of attack. This study sug- different thicknesses: NACA 0008, NACA 0010, and gests that computer costs can be reduced by limiting NACA 0012. Mach numbers for the NACA 0010 air- the number of transonic unstealjy aerodynamic calcu- foil were M = 0.76, 0.78, and 0.80. For the other two lations for small changes in airfoil geometry or angle of airfoils, the Mach number values were selected using attack.
the steady transonic similarity relationship of Liepmann and Roshko (1957). The NACA 0010 airfoil was used
Introduction
as the reference in the similarity relationship. Angle-of- attack effects were studied by including mean angle or Considerable research is being conducted presently attack in the XTRANZL calculations. Camber effects to develop finite-difference computer codes for cal- were examined by adding a simple parabolic camber culating transonic unsteady aerodynamics for a e r e distribution to the originally symmetric airfoils.
elastic applications. One such example is the two- dimensional finite-difference code XTRAN2L (Whit- low 1983), which solves the complete nonlinear small- disturbance potential equation for transonic flow. The airfoil semichord, c/2 XTRANZL code is a general-frequency version of the low-frequency LTRAN2 code of Ballhaus and Goorjian airfoil chord (1977). Houwink and Van der Vooren (1980) extended the range of applicability in the moderate-frequency lift coefficient due to plunge NLR version called LTR.AK2-NLR. The XTRAN21, lift coefficient due to pitch, rad-' code was developed by extensively modifying LTRAN2- NLR. For aeroelastic applications, finite-difference cal- pitching-moment coefficient about culations can become costly, especially if the flutter ana- quarter-chord due to plunge lyst is interested in determining aeroelastic characteris- tics for a wide range of airfoil geometry and mean angle pitching-moment coefficient abovt of attack. In order to limit the number of aerodynamic quarter-chord due to pitch, rad-' computations, the influence of changes in airfoil geom- etry and mean angle of attack on transonic unsteady pressure coefficient airloads needs to be understood.
critical preseure coefficient Airfoil shape and thickness effects on transonic airloads and flutter were initially studied by Bland scaied pressure coefficient (eq. (8)) and Edwards (1984). Calculations were made using XTRANZL for two airfoils at Mach numbers from 0.75 maximum camber height, nondi- to 0.83 in increments of 0.01. A conventional air- mensionalized by chord the algorithm by adding the time-derivative terms to value of d required to match he airfoil and wake boundary conditions. The resulting upper-surface steady shock loca- tion to that using 1.0" mean angle code was termed LTRAN2-NLR. The XTRAN2L code (Whitlow 1983), developed at NASA Langley Research of attack Center, is an extensive modification of LTRANZ-NLR scaled maximum camber height that solves the complete TSD equation and includes (es. (7)) monotone differencing, nonreflecting far-field boundary conditions, an improved grid, and a pulse-transient ca- plunge displacement in semichords h pability. Details of the XTRAN2L algorithm develop k reduced frequency, w b / U ment and modifications are given by Whitlow (1983).
Details of the grid development and pulse capability are free-stream Mach number rl M given bv Seidel, Bennett, and Whitlow (1983). All re- free-stream Mach number for M o o 10 sults of the present study were obtained using the time- NACA 0010 airfoil marching finite-difference computer code XTRANZL.
In the present study, pressure distributions and free-stream Mach numbers for generalized aerodynamic forces are computed for two study of thickness effects modes of airfoil motion: vertical translation (plunge) airfoil shape and pitch about the quarter-chord. Typically, unsteady aerodynamic forces are determined by calculating sev- time eral cycles of forced harmonic oscillation with the last free-stream velocity cycle providing the estimate oi the forces. Alternatively, harmoqic forces may be obtained indirectly from the re- streamwise coordinate relative to sponse due to a step change in a given mode of motion leading edge via the Duhamel integral (Ballhaus and Goorjian 1978).
coordinate normal to free stream, Although more economical, the step response method positive up leads to small errors resulting from the numerical a p proximation of the starting downwash (Seidel, Bennett, angle of attack, deg and Whitlow 1983). These errors are minimized in the mean angle of attack, deg present calculations by using a smoothly varying expo- nentially shaped pulse (Seidel, Bennett, and Whitlow scaled mean angle of attack 19E3). For pitch motion, the input pulse is given by (eq. (6)) ratio of specific heats maximum thickness-to-chord ratio and for pl mge motion, the input pulse is give^ by maximum thickness-to-chord ratio for NACA 0010 airfoil where AT is the nondimensional time step. The har- nondimensional time, U t l b monic response is obtained by dividing the Fourier size of nondimensional time step transform of the output force response by that of the input pulse. Use of the pulse-transient technique gives steady transonic similarity parame- considerable detail in the frequency domain with a sig- ter (eq. (4)) nificant reduction in cost over the alternative method steady transonic similarity parame- X I , X z , X 3 of calculating multiple harmonic responses. The accu- ters for study of thickness effects racy of the method for frequencies as high a~ k = 2 was demonstrated by Seidel, Bennett, and Whitlow (1983).
oscillation frequency, rad/sec Plunge and pitch pulse-transient calculations were performed using 2048 time steps with AT set equal to
Transonic Code XTRAN2L and Pulse-
5 ~ 1 3 2 , as done by Bland and EdwarLv (1984). Spuri-
Transient Technique
ous oscillations in the lift and moment coefficients de- The original LTRAN2 code (Ballhaus and Goorjian termined from the pulse-transient analysis sometimes 1977) was developed to time-accurately integrate the occur for reduced frequencies k 5 0.1. A typical exam- low-frequency transonic small-disturbance (TSD) equa- ple for the lift coefficient due to plunge q,, is shown in . 6 tion with steady-state airfoil and wake boundary con- figure l(a) for the NACA 0010 airfoil a t M = 0.78 and I ditions. Houwink and Van der Vooren (1980) improved a, = 1.0". The unsteady results are presented in the i form of real and imaginary coefficients as a function of In equation (3), the first term describes the airfoil reduced frequency k. The jagged nature of the curves at shape, the second term defines a simple parabolic cam- low frequencies generally occurs for airfoils with either ber line, and the third term represents the contribution angle of attack or camber at the higher Mach numbers. due to mean angle of attack about a quarter-chord pitch , In general, these oscillations are more severe for the axis. Airfoils described by the same function f(z/c) in lift coefficient than for the moment coefficient, and are equation (3) have similar steady flow fields when more severe for the airloads due to plunge than for the 1 - M 2 airloads due to pitch. These spurious oacillations were X = = Constant (4) traced to cases for which the lift and moment transient
[6 (7' + 1) ~ 2 1 ~ ' ~
responses did not return smoothly to their respective
where x is the steady transonic similarity parameter
steady-state values. Instead, for large values of T , they (Liepmann and Roshko 1957) and 7' = 2 - (2 - 7)M2.
tend to drift about these values with very small ampli- For a given value of X, the steady pressure distributions tudes (approximately two orders of magnitude smaller are identical when scaled according to Liepmann and than the maximum pulse response amplitude). Al- Roshko (1957) as follows: though this feature of the lift and moment time his- tories is hardly noticeable, the resulting low-frequency
C, [(T* + 1) MZ] 'I3
Fourier components are significant. This numerical dif- 6213 = Constant (5) ficulty was the result of information contained in the latter portion of the time histories and was alleviated It is clear from equation (3) that shape, camber, and by halving the number of time steps. This process also mean angle of attack may be varied independently. To beneficially reduces the total time by a factor of two.
maintain steady transonic similarity, however, these Figure l(b) shows clh results for the NACA 0010 airfoil variations must be done such that f(z/c) is the same calculated at the same conditions using 1024 time steps.
function.
The jagged low-frequency behavior of the clh curves has Shape effects were investigated by examining three been eliminated, and the pulse-transient computational airfoils of constant 6 but different shape S(z/c). Thick- time has been reduced by one-ha!!
ness effects were studied by assuming a shape S(z/c), Results of equal accuracy could be obtain.& with considering three different maximum thickness-to-chord an additional computational savings of approximately ratios 6, and adjusting a,/b or d/6 such that f(z/c) is 50 percent, by using a stepsize doubling procedure. In the same function. In both the shape and thickness this procedure, the time step size is doubled at times studies, effects due to mean angle of attack as well as corresponding to quarter intervals of the total time.
camber were independently investigated.
Time step sizes used are AT = 5 ~ 1 3 2 , 10x/32, 20~132, and 40x132, such that 480 time steps yield a total time
Linear Unsteady Airloads
that previously required 1024 time steps. Results ob- tained using stepsize aoubling for the NACA 0010 air- In this section, linear unsteady airloads are pre- foil are shown in figure l(c) and compare almost iden- sented for reference ar d for comparison with the tran- tically with the clh curves of figure l(b). Therefore, in sonic airloads which follow. The lift coefficient due this report, pulse cdculations for plunge motion were to plunge cl,, the pitching-moment coefficient due to recalculated for lifting cases (nonzero mean angle of at- plunge em,, the lift coefficient due to pitch cia, and the tack or camber) using stepsize doubling and 480 time pitching-moment coefficient due to pitch cma are shown steps to alleviate the low-frequency spurious effects.
in fi y r e s 2(a), 2(b), 2(c), and 2(d), respectively. These No further effort was expended to optimize the proce- reeults were computed using linear subsonic aerody- dure. Although the stepsize doubling procedure works namic theory at M = 0.80 (Bland 1982). All four aero- equally well for pitch motion, the pitch pulse results dynamic coefficients are smooth functions of reduced were not recalculirted, because the oscillations at low frequency k. Of particular interest are the moment co- values of k were generally not as severe in comparison efficients. For the pitching-moment coefficient due to with the plunge pulse rcsu1t.s.
plunge cmh (fig. 2(b)), the real part is a monotonically increasing function of k which is always positive; the General Airfoil Description .
imaginary part is a monotonically decreasing function The airfoil surt'ace may be expressed a8 of k. For the pitching-moment coefficient due to pitch cmp (fig. 2(d)), the real and imaginary parts are zero at k = 0, because the moments are calculated about the quart.er-chord (aerodynamic center). Both parts are al- ways positive and increase monotonically with reduced frequency.
Results for Three Airfoil Shapes
reduced frequencies, k > 0.3, the NACA 64A010 curves lie approximately halfway between the NACA 0010 and Shape effects were investigated by using three dif- parabolic-arc curves. For the pitching-moment coeffi- ferent symmetric airfoils, each with a 10-percent max- cient due to plunge c,, in figure 4(c), all three sets imum thickness-to-chord ratio (6 = 0.10). These of results again show similar trends with frequency. In three airfoil configurations, known as NACA 0010, contrast with the cl, comparisons of figure 4(b), agree- NACA 64A010, and parabolic arc, are shown in figure 3.
ment between the three sets of c,, results in figure 4(c) The NACA 0010 airfoil is defined by equation (6.2) of is not as good for low values of k. The difference8 also Abbott and Von Doenhoff (1959). Airfoil coordinates generally tend to become slightly larger with increasing for the NACA 64A010 airfoil were taken from the table Mach number. For the lift coefficient due to pitch cl, on page 356 of Abbott and Von Doenhoff (1959). For (fig. 4(d)), the results for the three airfoils are nearly P better definition of the NACA 64AO10 airfoil nose, identical throughout the entire range of reduced fre- three additional points on the upper and lower surfaces quency plotted. This suggests that cl, is relatively in- have been included by fitting an ellipse with the cor- dependent of shape for the three configurations consid- rect leading-edge radius to the leading edge and first ered. For the pitching-moment coefficient due to pitch ordinate. The NACA @ 0 1 0 and NACA 64A010 airfoils c , , ( f i ~ . 4(e)), large differences are observed between are very similar in shape (fig. 3), but the NACA 0010 the three sets of unsteady curves, which indicates there airfoil is slightly thicker than the NACA 64A010 w a r is a strong dependence upon shape, especially at low k the leading edge. The parabolic-arc airfoil has a very values. The dependence upon shape becomes larger at different shape with a sharp leading edge compared the higher Mb~i numbers. For example, the imaginary with the blnnt-nose NACA 0010 and NACA 64A010 part of c , , at M = 0.80 for the parabolic-arc airfoil airfoils. The maximum thickne~ses occur at 30 percent, has become negative for k < 0.12, which represents a 40 percent, and 55 percent chord for the NACA 0010, change in the obcillatory moment coefficient from lead- NACA 64A010, and parabolic-arc airfoils, respectively.
ing to lagging the harmonic pitch motion.
In the shape effects study, three cases are considered: For the three airfoils of different shape at a mean an- (1) Airfoils at a mean angle of attack of 0" with no gle of attack of 0' and with no camber, the unsteady air- camber; (2) Airfoils at a mean angle of attack of 1.0' loads show similar trends with frequency, even though with no camber; and (3) Airfoils at a mean angle of the steady pressure distributions and shock locations attack of 0" with camber. In case (3), the value assumed are very different. Comparing figures 4(b) through 4(e), for d is 0.00436. Pulse-transient calculations were then the lift coefficients shcw less of an effect due to air- performed using XTRAN2L for all three airfoils at foil shape than the moment coefficients. Differences M = 0.76, 0.78, and 0.80. The upper Mach number between unsteady forces, where they exist, are such limit was intentionally restricted in an attempt to avoid that the NACA 64AOlO forces generally lie approxi- the nonunique solutions arising from transonic small- mately halfway between the NACA 0010 and parabolic- disturbance theory as investigated by Williams, Bland, arc forces for a given value of k. This observation sug- and Edwards (1984) using XTRANZL.
gests that the differences in unsteady airloads for these airfoils may be related to differences between airfoil Mean Angle of Attack of 0" With No Camber maximum thickness locations or maximum C,, locations Steady pressure distributions for the NACA 0013, rather than to differences in airfoil shape.
NACA 64A010, and parabolic-,rc airfoils, all at a mean Mean Angle of Attack of 1.0" With No Camber angle of attack of 0' and with no camber, are shown in figure 4(a). At M = 0.76, the flow is subcritical Steady pressure distributions for the NACA 0010, for all three airfoils. At M = 0.78, small supersonic NACA 64A010, and parabolic-arc airfoils all at a, = regions have formed along both the upper and lower 1.0' and with no camber are shown in figure 5(a). At airfoil surfaces. At M = 0.80, shock waves of moderate M = 0.76, the NACA 0010 airfoil haa a supersonic re- s t r e ~ g t h are present on both surfaces at pproximately gion along the upper surface from approximately 5 per- 37 percent, 51 percent, and 65 percent chord for the cent to 30 percent chord which is terminated bv a weak NACA 0010, NACA 64A010, and parabolic-arc airfoils, shock wave. The NACA 64A010 and parabolic-arc air- respectively.
foils have insignificant upper-surface supersonic regions.
CI,, and c , , are Unsteady results for cl,, c , , , At Af = 0.78, there are shock waves on the upper sur- shown in figures 4(b), 4(c), 4(d), and 4(e), respectively.
faces of all three airfoils. At M = 0.80, the shocks For the lift coefficient due to plunge c ~ , in figure 4(b), are located farther downstream, and both the shock all three sets of results show similar trends as a function strengths and the size of the supersonic regions are in- of reduced frequency. The results for the three airfoils creased. Upper-surface steady shock locatione for each are virtually identical for low values of k. At higher of the three airfoils are very different. For the Mach number range considered, no shocks are present on the between the unsteady airloads for the three airfoils at lower surfaces of these airfoils with a, = 1.0'.
a, = 1.0" generally vary in the order of NACA 0010, NACA 64A010, and parabolic arc for the coefficients Unsteady results for cl,, C m h l cl,, and c , , are due to plunge (figs. 5(b) and 5(c)). For the coeffi- shown in figures 5(b), 5(c), 5(d), and 5(e), respec- tively. As shown in figure 5(b), the cl, curves for cients due to pitch (figs. 5(d) and 5(e)), this is not the three airfoils are very similar and almost coincide always the case. At M = 0.80 for example, the imagi- nary cia curves at low values of k for the NACA W10 for k < 0.3. For higher values of reduced frequency, the results for the similarly shaped NACA 0010 and and parabolic-arc airfoils are nearly the same, and the NACA 64A010 airfoils agree well, but the parabolic- NACA 64A010 values are no longer between these val- ues. At higher Mach numbers, the reversal in the or- arc results differ slightly. The agreement may be attributed to closer upper-surface steady shock loca- der of the results may be attributed to shock-wsociated tions for the NACA 0010 and NACA 64A010 airfoils, phenomena produced by including mean angle of at- tack. A comparison of unsteady airloads with a, = O0 which for M = 0.80 a r e at approximately 59 percent and a, = 1.0' (figs. 4 and 5, respectively) shows that and 63 percent chord, respectively. The upper-surface steady shock position for the parabolic-arc airfoil at the effect of mean angle of attack is greatest at the higher Mach numbers. Mean angle of attack also gen- M = 0.80 is near 78 perccnt chord. As shown in fig- erallv affects the moment coefficients more than the lift ure 5(c), the c , , results for the three airfoils are in good coefficient8 and affects the airloads due to pitch more general agreement. The curves for the NACA 0010 and than the airloads due to plunge.
NACA 64A010 airfoils again compare better with each other than with those of the parabolic arc, especially as Mach number is increased. The improved agreement at
Mean Angle of Attack of o0 With Camber
higher Mach numbers may be d.le to increasingly sim- ilar steady shock locations and steady shock strengths Steady pressure distributions for the N.iCA 0010, for the NACA 0010 and NACA 64A010 airfoils. Note NACA 64A010, and parabolic-arc airfoils a t a, = 0' that the real part of c , , has changed sign for k 5 0.3 and d = 0.00436 are shown in figure 6(a). Supersonic at M = 0.80. In figure 5(d), small oscillations in q, are regions are present along the upper surfaces of all evident at low k values as previously discussed. The clp three airfoils. These supersonic regions increase in size results for the three airfoils show excellent agreement for as the Mach number is increased and terminate with k > 0.1. In fact, at M = 0.76 and M = 0.78, the imag- strong shock waves. Steady s$ck locations for the inary cl, curves for the NACA 64A010 and parabolic- three airfoils are quite different. At M = 0.80, small arc airfoils coincide. This may be because of weaker supersonic regions occur on the lower surfaces of the steady shocks on the NACA 64A010 and parabolic-arc NACA 0010 and NACA 64A010 airfoils.
upper surfaces at M = 0.76 and M = 0.78 compared Unsteady results for the NACA 0010, NACA 64A010, with the NACA 0010 airfoil (fig. 5(a)). At M = 0.80, and parabolic-arc airfoils at a, = 0' and with camber the imaginary cl, curves in figure 5(d) for NACA 0010 are given in figures 6(b) through 6(e). As shown in and parabolic-arc airfoils coincide; the NACA 64A010 figure 6(b), co,nparimn of the cl, curves for the three curve has snaller negative values for low reduced fre- airfoils indicates a small effect of shape at higher re- quency. For the Mach numbers comidered, the q, re- duced frequencies. Differences between the unsteady sults shown in figure 5(d) indicate shape independence forces for the three airfoils occur in the same order ae ht, higher values of reduced frequency. In figure 5(e), the variation in airfoil maximum thickness location. At the c , , curves for the three airfoils show reasonable low k values, the cl, curves are nearly indistinguishable.
corre~ation for k > 0.3. For low k values, particularly As shown in figure 6(c), small differences exist between at M = 9.80, the unsteady XTRAN2L results are very the c,, coefficients which vary in a fashion similar to dependent upon shape. At M = 0.78, the NACA 0010 the variation of results in figure 6(b). At M = 0.80, and NACP. 64A010 curves are virtually identical for sign changes have occurred in the real part of c,, for all values cf k. At M = 0.80 and k < 0.1, the c , , low reduced frequencies, similar to the a, = 1.0" and curves vary in the order of NACA 64A010, NACA 0010, no camber results shown in figure 5(c). The el, coeffi- and parabolic arc. Also at M = 0.80, the imaginary cients for the three airfoils (fig. 6(d)) are essentially the part of cmo has changed aign for k < 0.2. These high same, except for the parabolic-arc airfoil at M = 0.80 Mach nurlber, low reduced-frequency effects may be at- and low k values. The NACA MA010 em, curveti are tributed to increased shock strengths and to farther-aft approximately halfway between the NACA 0010 and shock iocations due to nonzero mean angle of attack.
parabolic-arc curvea (fig. 6(e)). At M = 0.78, and ee- for the three airfoils of different shape at a, = 1.0' pecially at M = 0.80, the parabolic-arc results deviate and no camber, the unsteady aerodynamic coefficients from this spacing for low k values. This deviation may show similar trends with reduced frequency. Differences be attributed to a transonic effect of increased upper- surface shock strengths and to farther-aft shock loca- for a family of airfoils of different thickness, similarities tions caused by including camber. in unsteady airloads at Mach numbers which produce For the three airfoils of different shape at a, = 0' similar steady transonic flow fields.
and with a maximum camber height of 0.00436, the Figure 8 shows the steady transonic similarity pa- unsteady aerodynamic coefficients show similar charac-
rameter x as a function of Mach number for the three
teristics as a function of frequency. Differences between maximum thickness-to-chord ratios considered. Values unsteady airloads for the three airfoils vary smoothly for x are determined for 6 = 0.10 at M = 0.76,0.78, and and may be attributed to differences in maximum thick- 0.80 as listed in table 1. Also tabulated are the Mach ness location rather than airfoil shape. This may also numbers M I , Mz, and M3 for 6 = 0.08 and 6 = 0.12 be attributed to differences in maximum pressure loca- computed iterativ~ly using equation (4). Three cases tions or steady shock strekgths for the three airfoils. A Ere considered for investigation of the effects of thick- comparison of unsteady airloads at a, = 0' with no new, angle of attack, and camber as in the shape effects camber and with d = 0.00436 (figs. 4 and 6, respec- study. In each case, conditions are chosen such that the tively) shows that the effect of mean camber is amall at scaled steady pressure distributions on the three airfoils M = 0.76 and 0.78, and is larger at M = 0.80. This can are identical. Values for mean angle of attack a, and probably be attributed to increased upper-surface tran- maximum camber height d used in the thickness effects sonic effects produced by adding the parabolic camber study are listed in table 2. The values selected for mean line to the originally symmetric airfoils. As with the angle of attack a, result in a constant scaled mean angle mean angle of attack, camber also generally affects the of attack defined by moment coefficients more than the lift coefficients and the airloads due to pitch more than the airloads due to plunge.
In the shape effects study, unsteady forces for The values selected for maximum camber height d result the NACA 64A010 airfoil generally lie between the in a constant scaled maximum camber height defined by NACA 0010 and parabolic-arc unsteady forces. Dif- ferences in tranionic unsteady airloads for these airfoils may be related to airfoil maximum thickness location or maximum steady pressure location. The lift coeffi- cients show less of an effect due to airfoil shape than
Mean Angle of Attack of o0 With No Camber
the moment coefficients. Mean angle of attack or cam- ber generally affects the moment coefficients more than Steady pressure distributions for the NACA 0008, I the lift coefficients and the airloads due to pitch more NACA 0010, and NACA 0012 airfoils, ail a t a, = 0' than the airloads due to plunge. Also, comparison of and with no camber, are shown in figure 9(a). The the unsteady airloads (figs. 4 through 6) shows that the steady pressures are scaled to those of the reference effects due to mean angle of attack are similar to the NACA 0010 airfoil. Combining equations (4) and (5) effects due to camber. This is shown, for example, by leads to the simple scaling relationship comparing the M = 0.80 low reduced-frequency cma curves of figures 4(e), 5(e), and 6(e).
Results for Three Airfoil Thicknesses
As can be seen in figure 9(a), the scaled steady pres- Thickness effects were investigated by considering three symmetric airfoils of the same shape but with dif- sure distributions are identical at each Mach number ferent maximum thickness-to-chord ratios. All three (MI, Mz, and M3).
airfoils are defined by the shape expression, equa- Un~teady aerodynamic coefficients el,, c,, , c1, , and em, as functions of reduced frequency k are plot- tion (6.2) of Abbott and Von Doenhoff (1959), with a maximum thickness-to-chord ratio 6 of 0.08, 0.10, or ted in figures 9(b), 9(c), 9(d), and 9(e), respectively.
0.12 for the NACA 0008, NACA 0010, or NACA 0012 The unsteady aerodynamic coefficients are not scaled, because there are no known unsteady transonic simi- airfoils, respectively. These airfoil configurations are shown in figure 7. The study of the effects due to air- larity laws. For the lift coeficient due to plunge cl, shown in figure 9(b), the results for the three airfoils foil thickness was undertaken to investigate the agree are nearly coincident at low reduced frequencies. At ment in unsteady airloads for the NACA 64A010 and MBB-A3 airfoils (Bland and Edwards 1983) at slightly higher reduced frequencies, differences due to thichess are apparent. Here, the cl, results for the three airfoils different Mach numbers. These two airfoile have very of different thicknem do not show the consistent varia- s'milar shapes (S(z/c)) and differ primarily in thickness tions for a given k value that were found in the shape and camber. Therefore, it is of interest to investigate, effects ~ t u d y at a, = 0' and with no camber. For the the differences between the results for the th.-e air- pitching-moment coefficient due to plunge cmh shown foils become smaller. For the imaginary part of elh in figure 9(c), the three sets of curves again compare (fig. 10(b)) and the real part of c,, (fig. 10(c)), both well at low k values. As with the clh comparisons of at M = MI, the NACA 0010 curves are approximately figure 9(b), differences between the three sets of cmh re- midway between NACA 0008 and NACA 0012 curves.
sults occur at higher values of k. For both c;, and c m h , As Mach number is increased this characteristic deteri- these absolute differences are smallest a t the highest orates. In figure 10(d), the cr, results agree well for Mach numbers considered (M = Ad3). For the lift coef- the NACA 0008, NACA 0010, and NACA 0012 air- ficient due to pitch elo shown in figure 9(d), the three foils. This good agreement, especially at higher Mach sets of curves also indicate differences due to thickness. numbers, suggests that elm is relatively independent of At k = 0, the values for the real part of y o are slightly changes in thickness when 6, = 1.0'. In figure 10(e), different for the three airfoils, because the steady pres- the cmo curves for the lower Mach numbers exhibit the sure distributions, and hence lift coefficients, scale dif- same equally spaced characteristic as in the clh and c,, ferently than angle of attack. (See eqs. (6) and (8).) calculations. Of interest is the large increase with Mach For the pitching-moment coefficient due to pitch cma number in the real part of cma at k = 0. This change is (fig. 9(e)), the imaginary parts SLOW fairly close agree- a result of an aft movement of the aerodynamic center ment for low k values. The real part of cmo for the three produced by increasing Mach number.
airfoils shows large differences over most of the range of For the airfoils of different thickness at 6, = 1.0" reduced frequency plotted. and no camber, the unsteady forces have similar char- acteristics as a function of reduced frequency. Differ- For the NACA 0008, NACA 0010, and NACA 0012 airfoils, the unsteady airloads show similar trends with ences between the unsteady forces for the NACA 0008, NACA 0010, and NACA 0012 airfoils are apparent even frequency at the Mach numbers investigated. However, though the steady flow fields were scaled. Overall, the differences between the unsteady airloads for the three airfoils of different thickness are apparent, even though t,hree sets of results are in fairly good agreement. In general, the lift-coefficient results (figs. 10(b) and 10(d)) the steady flow fields are scaled. The differences may show better agreement than the moment coefficient re- be attributed to variation in maximum thickness or to differences in maximum steady pressure, because sults (figs. 10(c) and 10(e)). A comparison of unsteady airloads with and without a scaled mean angle of at- the maximum thickaess locations and maximum steady tack of 1.0" shows that the agreement between the un- pressure locations for the three airfoils are identical.
Agreement between unsteady results for these airfoils steady forces for the NACA 0008, NACA 0010, and NACA 0012 airfoils is improved with the inclusion of improves at the higher Mach numbers conside;ed. This is most clearly seen in the moment coefficients (figs. 9(c) scaled mean angle of attack. This is seen clearly, for and 9(e)), where the absolute differences between the example, by comparing differences in re~ults for the three airfoils for the pitching-moment coefficient due to three sets of unsh ady airloads decrease and occur at a pitch in figures 9(e) and 10(e). The improvement lower range of reduced frequency with increasing Mach number. These observations suggest that although in the agreemeat between the three sets of transonic un- steady airloads shown in figure 10(e) may be attributed effects due to thickness are important, effects due to to increased shock strengths and to farther-aft upper- differences in thickness become less important with increasing Mach number. surface shock locations produced by scaled mean angle of attack.
Scaled Mean Angle of Attack of 1.0' With No Mean Angle of Attack of 0" With Scaled Camber Camber Scaled steady pressure distributions for the NACA Scaled steady pressure distributions for the NACA 0008, NACA 0010, and NACA 0012 airfoils at a scaled 0008, NACA 0010, and NACA 0012 airfoils at 6 , = 0" mean angle of attack 6 , of 1.0" and with no camber and with a scaled parabolic camber of 0.00436 added are shown in figure lO(a). As before, the scaled steady to the originally symmetric profiles are shown in fig- pressure distributions for the three airfoils are the same ure ll(a). The scaled steady pressure distributions for for each of the three cases.
the three airfoils are the same for each of the three cases.
Unsteady results for the three airfoils a t 6 , = 1.0' Unsteady aerodynamic coefficients for the three air- and no camber are shows in figures 10(b) through 10(e). foils at 6, = 0" and with scaled camber are shown in In figure 10(b), the cl, cuwes for the NACA 0008, figures l l ( b ) through ll(e). As shown in figure ll(b), NACA 0010, and NACA 0012 airfoils are in good agree- the cl, curves for the NACA 0008, NACA 0010, and ment. In figure 10(c), the cmh results indicate a small NACA 0012 airfoils with scaled camber agree well, es- effect due to thickness. As Mach number increases, pecially at low reduced frequencies. The c,, results for the three airfoils also show good agreement at low k val- height do which match the steady shock locations with those using a , = 1.0'. In general, values for do are d e ues (fig. 11 (c)). Generally, absolute differences between the three sets of c , , curves get smaller as Mach number teimlned by making successive steady XTRANZL runs is increased. As shown in figure 1 l(d), the cl, curves in- and varying the maximum camber height d until the a , = 1.0' steady shock locations are matched. For dicate some effects caused by differences in thickness at the lower Mach numbers considered. Large differences both M = 0.76 and M = 0.78, the camber height was between the c,,, curves are observed for the three air- 0.00436 for the NACA 64A010 airfoil using XTRANZL.
foils (fig. ll(e)). This again demonstrates the effects For M= 0.80, the camber height is 0.0051.
Steady pressure distributions for the N 4 0 .. Clkt '
of thickness on the unsteady airloads. The differences become slightly smaller as Mach number is increased. airfoil, including either a , = 1.0' or . = d , , a', Also, the nonzero value for the real part of cmp at shown in figure 12(a). Upper-surface steady shock M = M3 and k = 0 is a result of the aft movement locations are at approximately 43 percent, 51. percent, of the aerodynamic center. and 63 percent chord for M = 0.76, 0.78, and G.80, For the airfoils of different thickne.3 at 6, = 0' and respectively. The two sets of steady pressure rerlults with d = 0.00436, the unsteady airloads have similar demonstrate that the shock locations are identical and trends with frequency. However, differences betweeu the shock strengths are similar. Also, the pressure the unsteady forces exist even when the steady pres- curves for the cambered airfoil show more of an aft sure distributions are scaled and the steady shock lo- loading than the pressure curvee for the airfoil at mean cations are matched. Overall, the unsteady results for angle of attack.
the three airfoils are in agreement, although the lift- Unsteady transonic airloads q,, c,,, cl, , and c , , coefficient results show less of an effect due to changes for the NACA 64A010 airfoi; are shown in figures 12(b), in thickness than the moment-coefficient results. Agree- 12(c), 12(d), and 12(e), respectively. For the lift co- ment between the three sets of results also generally im- efficient due to plunge (fig. 12(b)) and the pitching- proves as the Mach number is increased. A comparison moment coefficient due to plunge (fig. 12(c)), the a , = of unsteady airloads with no camber and with a scaled 1.0" results are nearly identical to the d = do camber camber of 0.00436 (figs. 9 and 11, respectively) shows results for the entire range of reduced frequency plot- that the agreement between the unsteady forces for the ted. For the lift coefficient due to pitch (fig. 12(d)), the three airfoil^ is slightly improved with the inclusion of two sets of results agree well. For the pitching-moment scaled camber. coefficient due to pitch (fig. 12(e)), larger differences In the thickness effects study, unsteady forces for betw-en the a , = 1.0' and d = do curves are observed the NACA 0008, NACA 0010, and SACA 0012 airfoils in comparison with the cl, , c,, , and el, results. The show that effects due to differences in thickness become largest difference between the cmo results for the two less important with increasing Mach number. Thc cases occurs in the imaginary part of c , , at M = 0.80 agreement between the unsteady forces for the three for k 5 0.25. These differences may be attributed to airfoils is improved with increasing Mach number, with differences in steady shock strengths.
the inclusion of scaled mean angle of attack, or with In general, comparison of the a, = 1.0" results with the inclusion of scaled camber. Also, comparison of the the d = do camber results shows good agreement. The unsteady airloads of figures 9 through 11 shows that the agreement is generally better for the coefficients due to effects due to scaled mean angle of attack are similar to plunge (figs. 12(b) and 12(c)) than for the coefficients the effects due to scaled camber. It is anticipated that if due to pitch (figs. 12(d) and 12(e)). This good agree- the steady shock locations and shock strengths were the ment suggests that for the same airfoil shape, combina- same, the transonic unsteady airloads for mean angle of tions d mean angle of attack and camber can result in attack and camber might compare even more favorably.
similar unsteady transonic airloads if the steady shock locations and shock etrengths are matched.
Comparison of Effects Due to Angle of
!
The results of this section, aa well aa results from .I
Attack With Effects Due to Camber
the shape and thickness studies, help explain the agree- Comparisons of unsteady results from both the ment between the harmonic aerodynamic forces of the shape and thickness studies show similarities between NACA 64A010 and MBB-A3 airfoils diecovered by effects due to mean angle of attack and camber. In this Bland and Edwards (1984). The agreement in unsteady section, effects due to mean angle of attack and effects forces between the NACA 64A010 airfoil at a , = 1.0' due to parabolic camber are directly compared for the and the MBB-A3 airfoil at a , = -0.5' occurred with a NACA 64A010 airfoil at M = 0.76, 0.78, and 0.80 by Mach number shift of 0.01, which held true for the en- matching steady shock locations. Angle-of-attack re- tire Mach number range considered, 0.75 < M < 0.80.
'I sults were computed uaing a , = 1.0'. Camber results The conventional airfoil used by Bland and Edwards were computed uaing values for the maximum camber (1984) wss the Amea model of the NACA MA010 airfoil I (Bland 1979). This airfoil differs from the symmetric The Its of this study give an indication of the 10-percent-thick NACA 64A010 airfoil considered here relative importance of small changes in airfoil configu- in that it is about 10.6 percent thick and has a very ration. Detailed comparisons of the transonic unsteady small amount of camber. The MBB-A3 is an 8.9 per- airloads aa functions of reduced frequency reveal simi- cent maximum thickness-to-chord airfoil with a super- larities in the results caused by changes in airfoil shape, critical camber distribution. As shown in figure 13, the thickness, camber, or angle of attack. Thew similari- shape (S(z/c)) of the MBB-A3 airfoil (without camber) ties offer insight into how the number of transonic un- compares doseiy with that of the NACA 64A010. Here, steady aerodynamic calculations might be limited for the NACA 64A010 airfoil was scaied down 11 percent small changes in airfoil geometry or mcan angle of at- for direct comparison with the MBB-A3. The shapes tack. This limitation results in a reduction in computer are very similar; therefore, the maximum thickness lo- costs.
cations are approximately equal. To determine waled At the same Mach number, the three airfoils with steady flow fields, a Mach number shift is required, be- different shapes (VACA 0010, NACA 64A010, and cause the two airfoils have different maximum thickness- parabolic arc) yield transonic ulisteady airloads that techord ratios 6. For the uncambered MBB-A3 airfoil have similar trends with reduced frequency, even though at M = 0.79, for example, the steady transonic similar- their steady pressure distributions and shock loca-
ity parameter x is 1.357. Holding the value for x fixed
tions are very different. Differences between unsteady to calculate the Mach number corresponding to the sim- aerodynamic forces, where they exist, are such that ilar steady flow field for the NACA 64A010 airfoil gives the NACA 64A010 forces are generally between the M = 0.776, or a Mach number shift of 0.014. With the NACA 0010 and parabolic-arc forces for a given value inclusion of camber for the MBB-A3 airfoil and mean of reduced frequency. The results show that the dif- angle of attack for both airfoils, the Mach number shift ferences in unsteady airloads for these airfoils are re- may be slightly different in order to match the steady lated to airfoil maximum thickness locations, maximum shock location. Actually, for similar steady flow fields, steady pressure locations, or steady shock strengths a mean angle of attack of 1.0" for the NACA 64AOlO rather than to differences in airfoil shape.
airfoil requires a mean angle of attack of 0.89" foi the At Mach numbers determined using steady tran- uncambered MBB-A3 airfoil. For the original MBB-A3 sonic similarity, the three airfoils of different thickness airfoil with positive camber, a mean angle of attack less (NACA 0008, NACA 0010, and NAC 4 0012) yitld trar- than 0.89" must be used to produce the same steady i
sonic unsteady airloads with simils - characteripf '
shock location. In the study by Biand and Edwards functions of reduced frequency. Agreement bets (1984), a mean angle of attack of -0.5" was arxumed ~ t e a d y results for the airfoils of differen? thicb 1- in all calculations for the MBB-A3 airfoil. Thm, by erally improves at the higher Mach numbers cc :d.
matching the steady shock locations and strengths for Agreement also improves by including (ucale aean two similar airfoils, the present results show that good angle of attack or by adding (scaled) camber to the orig- agreement between the unuteady transonic airloads ccn inally symmetric airfoil shape. The results show that, be expected. These results also explain the agreement although effects due to thickness are important, effects found by Bland and Edwards (1984).
due to differerrces in thickness become less important with increasing shock strength.
Concluding Remarks
In both the shape and thickness studies, similarities The effects of airfoil shape, thickness, camber, between effects due to mean angle of attack and effects and mean 'angle of attack on calculated transonic due to camber on transonic unsteady forcea were found.
unsteady airloads were investigated. Shape effects For the symmetric NACA 64A010 airfoil at a mean an- were atudied by considering three symmetric airfoils gle of attack of l.OO, the steady shock locations were with a 10-percent maximum thickness-to-chord ratio: matched with those of the same airfoil by including cam- NACA 0010, NACA 64A010, and parabolic arc. Thick- ber. The two configurations produced similar steady ness effects were investigated by considering three shock strengths and resulted in transonic unsteady air- symmetric airfoils of the same shape with differ- loads that showed very good agreement. Combinations ent maximum thickness-to-chord ratios: NACA 0008, of mean angle of attack a ~ d camber result in similar un- NACA 0010, and NACA 0012. Angle-of-attack and steady airloads if the steady shock locations and shock camber effects were studied in both the shape and thick- strengths are the same. The results of this study offer ness studies. The harmonic forces for airfoil plunge an explanation for the agreement in transonic unsteady and pitch motions were computed wing the pulse- airloads between the NACA MA010 and MBB-A3 air- transient technique of the XTRAN2L transonic small- foils reported by Bland and Edwards (J. Aitcr., vol. 21, disturbance code.
no. 3, Mar. 1984, pp. 209-217).
Bland, Samuel R.; and Edwardr, John W. 1984: Airfoil NASA Langley Reaevch Center Hampton, VA 23665 Shape and Thicknear Effects on Tranmnic Airloads and Flutter. J. A m r . , vol. 21, no. 3, Mar., pp. 204-217.
Novernoer 14, 1984 Houwink, R.; and Van der Vooren, J . 1960: Improved Ver- sion of LTRAN'L for Unsteady Tranmnic Flow Computa-
References
tions. AIAA J., vol. 18, no. 8, Aug., pp. 1008-1010.
Abbott, Ira H.; and Von Doenhoff, Albert E. c. 1959: Theory Liepmann, H. W.; and Rmhko, A, c.1957: Elements of Gwdynamtcs, John Wiley & Fws, Inc.
of Wtng Sections. Dover Publ., Inc.
Ballhaus, W. F.; and Goorjian, P. M. 1977: Implicit Vinite- Seidel, David A.; Bennett, Robert M.; and Whitlow, Difference Computations of Unsteady Transonic Flows Woodrow, Jr. 1983: An Ezploratory Study of Finitc- About Airfoils. AIAA J., vol. 15, no. 12, Dec., pp. 1728.- Daffercncc Grida for 7kamonic Umteady Aerodynamtcs. AIAA- 83-0503, Jan.
1735.
Ballhaus, W. F.; and Goorjian, P. M. 1978: Computation of Whitlow, Woodrow, Jr. 1983: XTRANgL: A Program Unsteady Transonic Flows by the Indicia1 Method. A'AA for Solving the General-Frequency Unsteady 'Itonsonic Small J., vol. 16, no. 2, Fcb., pp. 117-124. Dwturbonce Equataon. NAS.4 TM-85723.
Bland, S. R., compiler 1979: A GA RD Two-D!menstonal Williams, M. H.; Bland, S. R.; and Edwards, J . W. 1984: Acroclwtic Configurations. AGARD AR-156. Flow Instab:ltttes tn k m o n i c Small-Disturbance Theory.
NASA TM-86251.
Bland, Samuel R. 1982: Development of Low-Frequency Kernel-Functaon Aerodynamics for Comprwon Wtth Time- Dependent Fmtte-Dtffercncc Methods. NASA TM-83283.
TABLE 1. MACH NUMBERS FOR NACA 0008, NACA 3010, AND NACA 0012 AIRFOILS DETERMINED USING STEADY TRAONSONIC SIMILARITY Ml nirfoil 6 NACA 0008 0.08 0.7878 NACA 0010 .10 .76
/ NACA0012 .12 .7355 ,7569
TABLE 2. VALUES FOR MEAN ANGLE OF ATTACK AND MAXIMUM CAMBER HEIGHT USED IN THICKNESS EFFECTS STUDY
Airfoil I a,, deg I d
NACA 0008 i 0.8 0.00349
I
NACA 0010 .00436
I NACA 0 1 2 / ::: I .,524
NACA 0010
REDUCED FREQUENCY k (a) 2048 time steps.
4. 1 M=0.78
2.
REDUCED FREQUENCY k (b) 1024 time steps.
4. -
M=0.78
-
2.
0.0 .2 .4 .6 .8 1.0 REDUCED FREQUENCY k (c) Step-size doubling with 480 time steps.
Figure 1. Lift coefficient due to plunge el, for NACA 0010 airfoil at M = 0.78 and a, = 1.0'.
REDUCED FREQUENCY k REDUCED FREQUENCY k (a) Lift coefficient due to plunge.
(b) Pitching-moment coefficient due to plunge.
IMAGINARY REDUCED FREQUENCY k REDUCED FREQUENCY k (c) Lift coefficient due to pitch.
(d) Pitching-moment coefficient due to pitch.
Figure 2. Unsteady aerodynamic coefficients computed using eubeonic linear theory at A 4 = 0.80.
N A C A 0010
N A C A 6 4 A 0 1 0
PARABOLIC ARC
Figure 3. Profiles of NACA 0010,NACA 64A010, and parabolic-arc airfoils used in shape effects study.
N A C A 0 0 1 0
---
N A C A 6 4 A 0 1 0
----
PARABOLIC ARC (10%)
(a) Steady pressure distributions.
Figure 4. Results for airfoil shapes with no camber and cr, = 0".
NACA 0 0 1 0
---
NACA 6 4 A 0 1 0
----
PARABOLIC ARC
REDUCED FREQUENCY k
(b) Lift coefficient due to plunge.
Figure 4. Continued.
- NACA 0 0 1 0
----
NACA 6 4 A 0 1 0
C
I REAL
----- PARABOLIC ARC (10%)
I
REAL REAL
C
',mh
I I I I 1
-2. L
0.0 .2 .4 .6 .8 1 .O
REDUCED FREQUENCY k (c) Pitching-moment coefficient due to plunge.
Figure 4. Continued.
NACA 0 0 1 0
---
NACA 6 4 A O l O
----
PARABOLIC ARC (1 ox) -
"/ IMAGINARY REDUCED FREQUENCY k (d) Lift coefficient due to pitch.
Figure 4. Continued.
NACA 0 0 1 0
---
NACA 6 4 A 0 IMAGINARY
----
PARABOLIC ARC
REAL
r
L IMAGINARY
IMAGINARY REDUCED FREQUENCY (e) Pitching-moment coefficient due to pitch.
Figure 4. Concluded.
--
NACA 0 0 1 0
---
NACA 64A010
----
-----
PARABOLIC ARC (10%)
a = \ : - -- --
SURFACE SURFACE
1.2 1 I,,,,, 1.2
ir SURFACE (a) Steady p m e diitributions for airfoil upper and lower surfaces.
Figure 5. Results for airfoil shapes with no camber and a, = i.OO.
NACA 0 0 1 0
---
NACA 6 4 A 0 1 0
REAL
---- PARABOLIC ARC (10%) 1 : - 8.
0.0 .2 .4 .6 .8 1.0
REDUCED FREQUENCY k (b) Lift coefficient due to plunge.
Figure 5. Continued.
N A C A 0 0 1 0
---
N A C A 6 4 A 0 1 0
REAL
---- PARABOLIC ARC (10%)
IMAGINARY REAL REAL REDUCED FREQUENCY k (c) Pitching-momert caficient due to plunge.
Figure 5. Continued.
M = 0 . 8 0
N A C A 0 0 1 0
- - --
N A C A 6 4 A 0 1 0
REAL
---- PARABOLIC ARC (10%)
REAL +;EDUCED FREQUENCY k (d) Lift coefficient due to pitch.
Figure 5. Continued.
NACA 0010
---
NACA 6 4 A 0 1 0
1 IMAGINARY
----
PARABOLIC ARC (10%)
L IMAGINARY
IMAGINARY REAL
REDUCED FREQUENCY k
(e) Pitching-moment coefficient due to pitch.
Figure 5. Concluded.
NACA 0 0 1 0
NACA 6 4 A 0 1 0
PARABOLIC ARC (1 0%)
I \ REAL
-8.
0.0 .2 .4 .6 .8 1.0 REDUCED FREQUENCY k (b) Lift coefficient due to plunge.
Figure 6. Continued.
- N A C A 0 C 1 0
---
N A C A 6 4 A 0 1 0 L
REAL
----
PARABOLIC ARC (10%)
-
M = 0 . 7 8
I I I I I
F
t REAL
IMAGINARY 3.
REAL
C
mh
-
-1.
REDUCED FREQUENCY k (c) Pitching-moment coefficient due to plunge.
Figure 6. Continued.
NACA 0010
---
NACA 64A010
---- PARABOLIC ARC (10%)
REAL REAL
P-
-8. 9
0.0 .2 .4 .6 .8 1 .O
REDUCED FREQUENCY k (d) Lift coefficient due to pitch.
Figure 6. Continued.
N A C A 0010
---
N A C A 64A010 IMAGINARY
i IMAGINARY
2. - IMAGINARY
C
ma
REDUCED FREQUENCY k (e) Pitching-moment coefficient due to pitch.
Figure 6. Concluded.
N A C A
---
+ N A C A
N A C A
Figure 7. Profiles of NACA 0008, NACA 0010, and NACA 0012 airfoils used in thickness effects study.
STEADY TRANSONIC SIMILARITY PARAMETER x
NACA 0008
---
NACA 0010
----
NACA 0012
(a) Scaled steady pressure diatributiona.
Figure 9. Results for airfoil thickneaaecl with no camber and a, = OO.
M=M,
NACA 0008
---
NACA 0010
----
NACA 0012
REAL -2.
REDUCED FREQUENCY k (b) Lift coefficient due to plunge.
% Figure 9. Continued.
M=M,
NACA 0008
---
NACA 0010
REAL
----
NACA 0012
IMAGINARY \
I
REAL
I IMAGINARY 1
-- 0.0 .2 .4 .6 .8 1.0 REDUCED FREQUENCY k (c) Pitching-moment coefficient due to plunge.
Figure 9. Continued.
N A C A 0008
---
N A C A 0010
----
N A C A 0012
- w.
0.0 .2 .4 .6 .8 1.0 REDUCED FREQUENCY k (d) Liit coefficient due to pitch.
Figure 9. Continued.
N A C A 0008
--- N A C A 0010
L IMAGINARY
----
N A C A 0012
L IMAGINARY
IMAGINARY
C
mu
REDUCED FREQUENCY k (a) Pitching-moment coefficient due to pitch.
Figure 9. Conciuded.
N A C A 0008
---
N A C A 0010
.- - - - N A C A 0012
UPPER SURFACE
--------- -
LOWER SURFACE (a) Scaled steady pressure dbtributions.
Figure 10. Results for airfoil thicknesses with no camber and 5, = 1.0"
NACA 0008
---
NACA 0010
----
NACA 0012
r -
- REAL REDUCED FREQUENCY k (b; Lift coefficicnt due to plunge.
Figure 10. Continued.
N A C A 0008
---
N A C A 0010.
REAL
----
N A C A 001 2
I- REAL
REDUCED FREQUENCY k (c) Pitching-moment coefficient due to plunge.
Figure 10. Continued.
NACA 0008
---
NACA 0010
----
N A C A 001 2
REAL IMAGINARY
Y
REAL 4.
REDUCED FREQUENCY k (d) Lift coefficient due to pitch.
Figure 10. Continued.
i.'
C . , '
NACA 0008
---
NACA 0010
h IMAGINARY
----
NACA 0012
REDUCED FREQUENCY k (e) Pitching-moment coefficient due to pitch.
Figure 10. Concluded.
NACA 0008
---
NACA 0010
----
NACA 001:?
I,,,,,
UPPER
"1 SURFACE
(a) Scaled steady pressure distributions.
Figure 11. Results for airfoil thicknesses with 6 = 0.00436 and 6, = 0'.
--
NACA 0008
---
NACA 0010
----
NACA 001 2
REDUCED FREQUENCY k (b) Lift coefficient due to plunge.
Figure 11. Continued.
M=M,
NACA 0008
---
NACA 0010
REAL
i
----
NACA 0012
A=M, I IMAGINARY \
r REAL
REAL , I I I I 1 -2.
0.0 .2 .4 .6 .8 1.
REDUCED FREQUENCY k (c) Pitching-moment coefficient due to plunge.
Figure 11. Continued.
NACA 0008
---
NACA 0010
----
NACA 001 2
REAL
L L
IMAGINARY
Y
REDUCED FREQUENCY k (d) Lift coefficient due to pitch.
Figure 11. Continued.
NACA 0008
---
NACA 0010
----
NACA 0012
L IMAGINARY
IMAGINARY
-
REDUCED FREQUENCY k (e) Pitching-moment ccefficient due to pitch.
Figure 11. Concluded.
NACA 64A010
ao=l .o
---
d=do
.8 1 UPPER
SURFACE (a) Steady preaeure distributions.
Figure 12. Comparison of aerodynamic parameters for NACA MA010 airfoil with a, = 1.0' or d = do.
N A C A 64A010
a , = 1 .o
---
d=d,
I\, REAL
REDUCED FREQUENCY k (b) Lift coefficient due to plunge.
Figure 12. Continued.
NACA 64A010
M=0.80
a,= 1 .o
7-- d=d, REAL
78 j IMAGINARY \
1 REAL
REAL
REDUCED FREQUENCY k
(c) Pitching-moment coefficient due to plunge.
Figure 12. Continued.
NACA 64A010
a,= 1.0
---
d=d, IMAGINARY
Y
REAL
P
-8.
0.0 .2 . 4 .6 .8 1.0 REDUCED FREQUENCY k (d) Lift coefficient due to pi:& Figure 12. Continued.
NACA 64A010
a,=1.0
----
d=d,
IF IMAGINARY
L IMAGINARY
REDUCED FREQUENCY k (e) Pitching-moment coefficient due to pitch.
Figure 12. Concluded.
MBB-A3
---
MBB-A3 Wl THOUT CAMBER
----
N A C A 64A010 SCALED DOWN
-0.05 I-
Comparison of airfoil profiles of MBB-A3, MBB-A3 without camber, and scaled down NACA 64A010.
Figure 13.