Category Principles of Helicopter Aerodynamics Second Edition

With the advent of smart structures it is now becoming increasingly feasible to use smart compliant airfoil surfaces or several trailing edge mounted flaps on rotor blades

as a means of individually controlling the aerodynamic environment on each blade – see Section 6.12. Actively controlling the blade airloads as a function of blade azimuth position offers tremendous possibilities in improving helicopter rotor performance, as well as re­ducing blade loads and vibration levels and perhaps even reducing rotor noise. See Lorber (2000) and Friedmann (2004) for a good summary of what may be possible. However, prac­tical concern of active blade adaptation through the application of flaps, airfoil camber or other chordwise shape changes is the availability of suitable low mass, high force actuators. These actuators must also be mounted inside the rotating blade and must be used to drive the aerodynamic surfaces at relatively high physical and reduced frequencies.

Theoretical studies of these types of moving flap problems using advanced helicopter rotor models require the use of a suitably formulated time-domain theory for the blade section aerodynamics. An unsteady aerodynamic theory is required, first because the local camber actuation frequency may be several multiples of the rotor rotational frequency and, second, because high resolution predictions of both the unsteady airloads and acoustics need to be made. In addition, because the local effective reduced frequencies based on active camber motion may become large, incompressible assumptions may not be adequate despite the exact analytical neatness of the theory.

As previously discussed in Section 8.13, for a rotor in forward flight a blade element will encounter a time-varying incident velocity and Mach number. Under these conditions, there are additional unsteady aerodynamic effects to be considered, including the effects of the nonuniform convection of the shed wake (see Fig. 8.25) as well as noncir – culatory effects. If compressibility effects are to be included, the mathematical modeling of the problem becomes relatively difficult because, unlike the incompressible flow case, no exact solution can be found. However, extending the compressible indicial theory from the previous sections it can be shown that for nonsteady variations in Mach number at a constant AoA a that for the circulatory part of the unsteady lift

{o)fc{s-o)do, (8.199)

and for the noncirculatory part of the unsteady lift

(8.200)

where Mo is the initial Mach number and all the symbols have their usual meanings as defined in the preceding sections. See Jose & Leishman (2005) for details.

Representative calculated results are shown in Fig. 8.40 for time-dependent free-stream Mach number variations with the airfoil at a constant angle of attack, and are compared with direct CFD calculations based on an Euler method. Notice again the applicability of the indicial method for this type of calculation, which is several orders of magnitude more computationally efficient than the CFD solution, and so is ideal for various forms of helicopter rotor analyses where the treatment of these unsteady effects is necessary. It will be apparent that the unsteady incompressible flow theory (see Section 8.13) fails for these conditions, again emphasizing that the proper treatment of compressibility effects will be necessary if accurate unsteady airloads predictions are to be obtained on the rotor.

In the subsonic case, the time-varying lift and pitching moment during the penetration of a convecting sharp-edged gust can only be found by numerical means. The problem, however, can be simplified if it is approached by using the reverse flow theorems, as discussed previously; see also Leishman (1997). Results for the lift on the airfoil penetrating a traveling sharp-edged gust at a Mach number of 0.5 are shown in Fig. 8.38 for various gust speed ratios. Results at other Mach numbers are qualitatively similar. Notice that the effect of increasing the gust convection velocity is to increase the rate of buildup of lift, analogous to the incompressible case shown previously in Figs. 8.23 and 8.24. For the subsonic case the center of pressure is always initially forward of the 1 /4-chord as the airfoil penetrates into the gust front, but it moves back quickly again after the airfoil becomes fully immersed in the gust. After only a short time, the aerodynamic center is fully at the 1 /4-chord.

Lift and pitching moment results for convecting sharp-edged gusts in subsonic flow have also been computed by Singh & Baeder (1997b) using an Euler CFD method. Representative

results for the lift are shown in Fig. 8.38 for a Mach number of 0.5 and for several gust speed ratios. Notice that in the early period where s < 2M/(1 + M), or 0.67 semi-chords at this Mach number, the lift varies almost linearly with time as predicted by the exact subsonic linear theory, the rate of growth increasing with increasing gust convection velocity. The comparisons are excellent and lend significant credibility to the CFD results, which can provide valuable solutions for later values of time where exact analytic solutions are not possible. Like the incompressible results, the CFD results predict an initial lift overshoot for s > 2M/(1 + M) that reaches a peak when the airfoil is about halfway into the gust.

To use the indieial method to examine arbitrary convecting vertical gust problems, the convecting sharp-edged gust solutions must be approximated by exponential functions. One suitable exponential approximation to the lift produced on an airfoil encountering a convecting sharp-edged gust is

Ci{s)

w0/V

where all the coefficients will, in general, be Mach number dependent. To satisfy the initial conditions at 5 = 0 then <7/ = 1- Also, g,- > 0 for і = 1,…, N + 2. The transient shown in the lift response at small values of time for fast traveling gusts is represented by the second two terms in Eq. 8.198, where the coefficient GN+i and the differences in the values of the time constants g#+1 and gw+2 will affect the size and width of this transient. Physically, this transient is a result of the accumulation of pressure waves. In the limit when A -» 0, the magnitude of the transient approaches the piston theory value of 4/М. Results of this procedure are shown in Fig. 8.38 for several gust speed ratios at a Mach number of 0.5. It is seen that while an exact fit to the initial transient at smaller values of A cannot be obtained, an acceptable level of accuracy is possible for values of A that are not too far from unity.

The problem of convecting vortices on the unsteady lift and noise generation has been addressed by Singh & Baeder (1997b) and Leishnian (1997). Results for the unsteady lift on the airfoil for the same conditions of Fig. 8.37 are shown in Fig. 8.39 for several gust speed

Distance-traveled in semi-chords, s

ratios. The results are all referenced to the к = 1 case, so that for downstream traveling vortices the ВVI encounter occurs progressively earlier in time (or distance). We see that an increase in vortex convection speed (decrease in k) progressively increases the peak-to – peak value of the unsteady lift, but more importantly, increases the time rate of change of lift. This will be reflected in the corresponding acoustic field, where the BVI sound pulse (see Fig. 8.48) will increase significantly in magnitude even for values of к not too much lower than unity. Figure 8.39 also shows results for the BVI problem as computed directly using the Euler СГО method; the correlation of the indicial results with the Euler solution is excellent. Overall, these results confirm that the gust speed ratio of the tip vortices in the rotor wake will be a necessary parameter to account for in helicopter blade unsteady airloads and aeroacoustics analyses.

Blade-vortex interaction (BVI) is one practical example of a helicopter flow with an intense vertical velocity field with high velocity gradients. The 2-D BVI problem, which is shown schematically in Fig. 8.36, has been widely addressed in the literature – see, for example, McCroskey & Goorjian (1983) and Singh & Baeder (1997b). While passing an airfoil at a predetermined distance, a convecting vortex of positive circulation produces a downwash velocity while upstream of the blade (airfoil), and this changes to an upwash as it moves downstream. This situation leads to a rapidly and continuously changing angle of attack, resulting in highly unsteady aerodynamic lift and other airloads. The acoustic pressure (or noise) propagated to an observer from such a blade-vortex encounter is related, as given by the compact source limit, to the time rate of change of this lift (see Section 8.19).

Representative numerical calculations will now be shown for the unsteady loads on a NACA 0012 airfoil interacting with a convecting vortex of nondimensional strength Г = Г/(Ус) = 0.2. The airfoil is assumed to have zero angle of attack. The interacting vortex was assumed to have a Lamb-like normal velocity distribution (see Eq. 10.9) given

(8.197)

with r2 = (x — xv)2 – f (y — yv)2, where xv, yv refer to the position of the vortex relative to a coordinate axis at the leading edge of the airfoil. The reciprocal influence of the airfoil on the vortex convection velocity and trajectory is neglected, and we can assume that the vortex remains undisturbed from the path yv = yQ = —0.26. As shown by Srinivasan& McCroskey

(1987) , the effect of vortex trajectory distortion appears to be of secondary importance to the overall airloads except for the transonic case when the vortex may pass close to a shock wave. A viscous core of dimensions rc = 0.05c was used for the calculations, although the interaction between the airfoil and the vortex is sufficiently spaced in this case that the core radius does not play a role.

Results for Mach numbers between 0.65 and 0.8 are shown in Fig. 8.37. The results are compared to a CFD solution based on the Euler equations. Notice the good agreement between the indicial approach and the CFD solution, the results essentially confirming the validity of linear theory for this problem. Even for the higher Mach number of 0.8, where some nonlinearities because of the transonic nature of the flow might be expected, the agreement is good, although there is a somewhat larger lift overshoot downstream of the airfoil trailing edge compared to that predicted by the Euler method. The results in Fig. 8.37 also indicate that that the peak-to-peak value of the lift response becomes attenuated with increasing Mach number, which is exactly the opposite to the result obtained using quasi­steady, subsonic airfoil theory or incompressible unsteady airfoil theory. Also, it is apparent that the effects of increasing Mach number introduces a significantly larger phase lag in the lift response, which obviously becomes a critical consideration when accurate noise predictions are an issue (see also. Question 8.12).

If the subsonic sharp-edged gust functions are approximated in exponential form, the same techniques used previously to find the total unsteady lift to an arbitrary forcing can be applied. Within the assumptions of the linear theory, a general stationary vertical gust field, wg(x, t), can be decomposed into a series of sharp-edged vertical gusts of small magnitude. When the approximation to the aerodynamic response for sharp-edged vertical gust is found, then the response to an arbitrary vertical gust field can be found using linear superposition by means of Duhamel’s integral. After the initial transient dies out, the response to a continuous vertical gust field may be written analytically as

ЛС*(s) = ^ jf ^-^(s – a, M)da^. (8.191)

As described previously, the Duhamel superposition can be performed numerically in various ways, including the state-space (continuous time) form or the one-step recursive formulation (discrete time) form. In the latter case, a finite-difference approximation to the Duhamel integral leads to a solution for the lift that may be constructed from an accumulating series of small vertical gust inputs using

ACf (0 = j-i [ wg(s) – Z,(,) – Z2(i)] , (8.192)

where the terms Z and Z2 are the deficiency functions. In this case, using Algorithm D-2 given previously in Section 8.14.1, the deficiency functions are given by the one-step recursive formulas

Z(s) = Z(s — As)E + G[lUgC?) — wg(s — As)]#^, (8.193)

Z2CS) = Zt{s — £ss)E2 + G2[u;g(s) — wg(s — A, s)]E2/2, (8.194)

where E1 = exp(—g f2As) and E2 = exp(—g2d2As).

By applying Laplace transforms to the exponential approximation to the sharp-edged vertical gust function in Eq. 8.188, the lift transfer function relating the output (the lift) to the input (the vertical gust field) can be obtained. From the transfer function, the alternative state-space form of the equations can be written as

Ui(Ol_r 0 1 lfzi(Ol

l^(OJ _-glg2{^r)2 P* ~(g + g2) (^) P2 J і Z2(0 I

| 1 0 j AWg(t)

a Lift

with the corresponding output equation for the total normal force (lift) coefficient for the arbitrary vertical gust field as

These equations can then be solved using a standard ordinary differential equation solver for any arbitrarily imposed vertical gust field.

Mazelsky (1952a) and Mazelsky & Drishler (1952) have obtained exponential ap­proximations to the stationary sharp-edged vertical gust functions at various Mach numbers by means of reciprocal theorems in conjunction with numerical solutions for airloads com­puted in the frequency domain. A suitable functional approximation for the sharp-edged vertical gust is of the form

N

ir(s, Gie-8i (8.186)

;=i

where the Gi and gi coefficients will all be Mach number dependent, and with j Gі = 1

and gi, і = 1,…, N > 0. The corresponding lift is given by

(8.187)

It has been shown previously that the circulatory part of the total lift from a step change in AoA in subsonic compressible flow can be approximated by a two-term exponential func­tion, and for all subsonic Mach numbers the results are related through a characteristic time
that can be scaled in terms of Mach number alone. For later values of time the sharp-edged vertical gust and indicial AoA functions must approach each other; thus it is reasonable also to assume a similar behavior for the vertical gust function. It can be approximated by an exponential function of the form

N N

і £g, = i, gi> o,

i=l i=l but now the G s and gs are fixed and considered independent of Mach number. Like the

oi ЛлЛ _ __________ —_____ и~______ —и~

іиисаіиїла given рісуіииму, ьиеи an ajjpnjAimaiiuii eon uc аьмииси їй uc

valid up to at least the critical Mach number of the airfoil, beyond which nonlinear effects associated with transonic flow do not allow for such relatively simple generalizations.

The solution for the coefficients in Eq. 8.188 can now be formulated as a least-squares optimization problem with several imposed constraints. In this case, however, there are no equivalent experimental results for airfoils in sinusoidal gust fields. Therefore, direct time domain results obtained using analytic and CFD solutions can be used as a reference to find the gust function approximations in exponential form. To obtain an approximation to the exact linear theory, one constraint can be imposed by matching the time rate of change of the exact solution (Eq. 8.183) and approximate solution (Eq. 8.187) at s = 0. This helps constrain the solution to ensure that the exact result will always be closely obtained in the initial stages. This part of the response is particularly important for transient aerodynamic phenomena such as BVI – see Section 8.16.4. Differentiating Eqs. 8.183 and 8.188 with respect to s and equating their gradients at s = 0 leads to a definition for the first constraint, namely

This result cannot be obtained over the entire subsonic Mach number regime; however, an evaluation of right-hand side of Eq. 8.189 shows that it is numerically close to 0.6 over the practical range 0.2 < M < 0.8. As M -»■ 0, the slope tends to infinity at s = 0, which is consistent with the exact solution given by von Karman & Sears (1938). In the latter case, with any common type of exponential approximation to the exact incompressible solution [see, for example, Bisplinghoff et al. (1955)] the gradients cannot be matched as s -* 0. In addition, a constraint is imposed for the initial conditions, namely Y^=i G, — 1 = 0, and also Gj, gi, > 0, і = 1, 2,…, N. Also, as 5 сю, the airloads approach the value given by the usual steady-state subsonic linearized airfoil theory [i. e., C„(s = оо, M) = 2п/ft]. In a parallel way to that described previously for the indicial AoA case, a 2A-dimensional vector of unknown coefficients can now be defined as

хГ = {Gi &2 • • • g g2… gw) •

The vector in Eq. 8.190 can be chosen to minimize the differences between the approxi­mating exponential vertical gust function and the exact or any reference solutions over the domain of s and M.

Exact results for the vertical gust response in linearized subsonic flow can be com­puted using the solutions given by Lomax (1968) and Lomax et al. (1952), but only up to s = 4M/(1 — M2). For higher Mach numbers this corresponds to finding an exact solution up to about 10 semi-chord lengths of airfoil travel. CFD results for the sharp-edged gust

20

problem have been made available by Singh & Baeder (1997b). Notice that the exponential approximations to the gust response, which are shown in Fig. 8.35, match the exact solutions and the CFD solutions almost precisely. It is apparent that although the final values increase with increasing Mach number, the initial growth in lift is less.

The resulting coefficients for the indicial gust functions are given in Table 8.3. Results for the N = 3 case at M = 0.5 as given by Mazelsky & Drischler (1952) are also tabulated. The coefficients of the generalized subsonic vertical gust function as M 0 are close to those given by R. T. Jones (1938,1940) for the incompressible case (the N = 2 exponential approximation to the Kussner function) and confirms that the results for the subsonic case are closely approximated by scaling the g coefficients by /32; that is, the aerodynamic vertical gust responses are related in subsonic flow, albeit approximately, through a characteristic time.

In the rotor plane, there are a large number of vortical disturbances that lie in proximity to the blades. This is especially significant on the advancing and retreating sides of the rotor where the blades may interact with tip vortices – see Section 8.16.4. The unsteady forces produced on a rotor blade arise primarily because of the vertical velocity between the wake disturbance (gust field) and the airfoil surface. In linear theory, this is treated as an imposed unsteady upwash field, which must be used to satisfy the boundary conditions of flow tangency on the airfoil surface. As described previously, within the assumptions of linear theory, incompressible flow solutions for the sinusoidal vertical gust problem have been solved by Sears (1940), and exact solutions for the sharp-edged vertical gust problem have been found by Kiissner (1935), von Karman & Sears (1938), and Miles (1956).

8.16.1 Exact Subsonic Linear Theory

For the subsonic compressible flow case, the problem of finding the sharp-edged vertical gust response, тf/(s, M), was considered by Lomax (1953) using a similar approach to that described previously to derive the indicial responses from changes in airfoil AoA and pitch rate. The subsonic vertical gust result was also obtained by Heaslet & Sprieter (1952) by means of reciprocal relations. The actual mathematical calculations are fairly involved, but Lomax (1953) has shown that exact analytical expressions for the airfoil pressure distribution can be found for a limited period of time after the vertical gust entry. For the period 0 < s < 2M/(1 + M) the lift coefficient is given by

(8.183)

for a gust velocity perturbation of Awg. me corresponding 1/4-chord pitching moment coefficient for the period 0 < s < 2M/(1 + M) can also be found, giving

(8.184)

Although these results are valid for less than one-chord length of airfoil travel for all subsonic Mach numbers, these analytic solutions are exact within the underlying assumptions of linearized, subsonic, unsteady thin-airfoil theory.

One interesting result from Eq. 8.183 is that increasing Mach number decreases the initial rate of lift production for a given distance traveled during the vertical gust penetration, perhaps not an intuitive result. However, a similar result has been shown previously for the indicial AoA case, where there is an increasing lag in the development of the circulatory lift for higher subsonic Mach numbers – see also Bisplinghoff et al. (1955). Using Eq. 8.183 it can be shown that the lift builds very rapidly during the vertical gust penetration, reaching close to one third of its final value (2 л ffi) shortly after the airfoil becomes fully immersed in the vertical gust (that is, when s = 2).

The position of the center of pressure during the vertical gust penetration is also of interest. Heiicoptef rotor blades tend to be relatively compliant in torsion compared to the wings of fixed-wing aircraft, and so the variation in airfoil pitching moment about the elastic axis can be very important. For the period 0 < s < 2M/(1 + M), the center of pressure can be computed from Eqs. 8.183 and 8.184 giving

which shows that xcp moves quickly aft to the 1/4-chord location within the short period 5 = 2M/(1 4- M). Therefore, for most practical purposes it is sufficient to assume that the aerodynamic center remains at the 1/4-chord throughout the vertical gust penetration. This result is also consistent with the incompressible solution of von Karman & Sears (1938). For later values of time up to s = 4M/(1 — M2), solutions for the airfoil pressure distribution during the vertical gust penetration takes a more complicated form, and the determination of the lift and pitching moment is only possible by means of numerical methods. For s > 4M/(1 — M2) no exact solutions to the sharp-edged vertical gust problem are possible in subsonic flow by means of the linear theory; consequently other and usually more approximate numerical methods or CFD solutions must be adopted.

Whereas the formulation of the subsonic unsteady aerodynamic model in the previ­ous section has been derived, in part, by using experimental measurements in the frequency domain, the validity of the model must be reconfirmed by comparing with experimental results of unsteady forces and pitching moments as functions of time or angle of attack. The unsteady airloads in response to an arbitrary time history of a and q can be obtained by using the previously derived indicial functions with Duhamel superposition in the form of the recurrence equations, or by integration of the state-space equations. Results will now be shown for both a pure plunging oscillation and also for a pitching oscillation; the absence of pitch rate in the plunge oscillations makes it possible to isolate the effects of the pitch rate in the unsteady aerodynamic response. For the plunging case, an equivalent

Angle of attack, a (deg.)

quasi-steady AoA can be defined using

Notice that, unlike the pitch angle, the equivalent AoA in Eq. 8.182 is not a directly mea­surable quantity because it depends on the free stream velocity, V.

Representative variations in normal (lift) force and 1 /4-chord pitching moment coeffi­cients are shown in Fig. 8.32 for harmonic plunge forcing and in Fig. 8.33 for harmonic pitch forcing. In each case, the results obtained from the subsonic unsteady aerodynamic model are compared to experimental measurements made by Liiva et al. (1968). The pitch and plunge data have been selected for approximately the same reduced frequency and mean AoA, although the equivalent amplitude of forcing is slightly different. The results show the expected characteristic elliptical shaped normal force and pitching moment loops symptomatic of attached flow. It is clear that in both cases there is a reduction of the un­steady “lift slopes” compared to the steady case, but with a larger phase lag (width of the hysteresis loop) for the plunging case. The width (amplitude) of the moment loop for the pitching oscillation is much larger than that for the plunge oscillation because of the

pitch rate contributions.[36] For each mode of forcing, it is apparent that the major axis of the unsteady pitching moment loop is closely aligned with the steady moment slope. This positive slope is a result of the aerodynamic center being slightly forward of the 1 /4-chord measurement axis.

Further results showing quality of the predictions of the unsteady lift in compressible flow and the improvement that can be obtained over Theodorsen’s theory are given in Fig. 8.34 for pitching oscillations. To summarize as many operating conditions as possible, the results are shown in terms of lift and pitching moment amplitude and corresponding phase angle versus reduced frequency. To put the measurements on a common basis, the amplitudes of the response have been expressed as ratios by normalizing with respect to the measured static lift curve slope. Plunge oscillation data are more scarce, but as mentioned previously, they are particularly useful because of the absence of loading contributions from pitch rate (q) terms. It is clear from the results obtained in Fig. 8.34 that there is a good
correlation between the predicted and measured lift response. This provides considerable support for using this type of unsteady aerodynamic modeling over the Mach number and reduced frequency range appropriate for helicopter applications.

The subsonic compressible flow theory compares more favorably with measurements than Theodorsen’s theory, particularly for the phase. The pitching moment is, however, more difficult to predict without the use of empirical results. Notice from Fig. 8.34(d) that there is a gradual divergence of the phase angle as the reduced frequency tends to zero. As described previously in Section 7.7.1, this is attributable to an offset of the aerodynamic center from the 1 /4-chord moment reference point, and it emphasizes the need to carefully represent the changing aerodynamic center with Mach number in a rotor simulation if accurate predictions of the unsteady aerodynamic effects are to be obtained.

By suitably generalizing the indicial response in terms of exponential functions and Mach number as shown previously, the corresponding state-space realization may also be obtained for each component of the loading in a subsonic compressible flow – see Leishman & Nguyen (1990). Firstly, consider the normal force response to continuous forcing in terms of angle of attack. Using the indicial functions given previously, the circulatory normal force response to a variation in AoA can be written in state-space form as

(8.175)

with the output equation for the normal force coefficient given by

where Ъх/fi is the lift-curve-slope for linearized compressible flow and 0:3/4 is the AoA at the 3/4-chord, that is,

о:з/4(0 = a(t) + —.

Similarly, the noncirculatory normal force from AoA can be written in the state-space representation as

x3 = a(t)——— !— x3 = o:(0 + «33X3, (8.177)

каТі

with the output equation for the normal force coefficient given by

C"‘(0 = 4;,. (8.178)

The remaining state equations for the pitching moment and pitch rate terms can be derived in a similar way using all of the other indicial response approximations for pitching moments and pitch rate, as given previously. The individual components of aerodynamic loading are then linearly combined to obtain the overall aerodynamic response. For example, the total normal force coefficient is given by

Cn{t) = Ccn{t) + Cnny) + Cnnc(t)

and an analogous equation holds for the pitching moment about the 1/4-chord. Thus, the overall unsteady aerodynamic response can be described in terms of a two-input, two-output system where the inputs are the airfoil AoA and pitch rate and the outputs are the unsteady normal force (lift) and pitching moment. It can be shown that by rearranging the state equations, the inputs and outputs can be represented in the general form

which involves a bilinear combination of the states xi and X2. Thus, as a byproduct of the above system representation for the unsteady lift, the necessary information may be extracted from the system at a given instant of time to obtain the unsteady axial force component. Finally, the instantaneous pressure drag can be obtained by resolving the components of the normal force and chordwise forces through the geometric AoA a using Eq. 8.174.

Helicopter rotor blades have a much lower stiffness and effective damping than the wing of a fixed wing aircraft for the in-plane (lead-lag) degree of freedom. Whereas the blade flapping and torsion degrees of freedom are primarily influenced by the lift and pitching moment respectively, the lead-lag degree of freedom is influenced by the drag. Furthermore, the blade lead-lag motion may couple with the flapping or torsion degrees of freedom and may lead to an aeroelastic instability of the blades. For example, this is intrinsic to the problem of ground resonance discussed in Section 4.10. As shown in Chapter 4, these coupling effects result from both the Coriolis forces and the aerodynamic loads. Therefore, for a comprehensive model of the rotor system it is necessary to include an accurate representation of the unsteady aerodynamic loads for all three degrees of freedom (lift, pitching moment, and drag).

For steady flow conditions, the pressure drag coefficient Cdp may be computed by re­solving the normal force and axial force (or the leading edge suction force) coefficients through the AoA a using

Cdp = Cn sin a — Cacos a, (8.163)

as shown in Section 7.11.1. For steady flow the normal force coefficient is given in terms of the force curve slope, C„a(M), at a given Mach number, M, and the angle of attack, a, as

Cn = СПа{М)а. (8.164)

The corresponding axial (chord) force coefficient for these conditions is given by

Ca = СПа(М)и tana. (8.165)

where a is measured in radians. Now, it is well known that for steady potential flow the pressure drag, Cdp, is identically zero (d’Alembert’s paradox), that is,

Cdp = Cn sina—Ca cosa = C„a(M)a sina—СПа(М)а tana cosa = 0. (8.166)

However, for a real flow there is a net pressure drag on the airfoil because of viscous effects on the chordwise pressure distribution. As shown Section 7.11.1, the inability of the airfoil to attain 100% leading edge suction is modeled using the recovery factor r}a such that

Ca = Ца Cna(M)ct tana, (8.167)

where the value of rja may be adjusted empirically as necessary to give the best fit with the static axial force and/or drag measurements. Typically, the value of r)a is approximately 0.95.

The viscous (shear stress) drag is represented by the term Cj0 and is also a function of Mach number. However, its value is nominally constant for the AoA range below stall. Thus, the total drag in steady flow is given by the sum of the pressure and viscous shear components as:

Cd = Cd0 + cdp = Cdo + cn sina – Ca cosa. (8.168)

The unsteady axial force coefficient, Ca(t), depends only on the circulatory component of the loading. This is easily proved by considering the chordwise form of the circulatory and noncirculatory pressure loadings. The circulatory form is given by the standard thin-airfoil theory result, namely

The important point here is that the circulatory form has a leading edge pressure singularity, and this form of pressure distribution is obtained no matter what the value of the effective angle of attack, ae. The noncirculatory form (at time zero) is given by the piston theory result

ACpC(x, t = 0) = —a.

While this initial loading changes with time as pressure waves propagate from the airfoil, no leading edge singularity exists for any time. The general expression for the leading edge suction force is

Са = П – lim [AClx}, (8.171)

o Jc-»0 и

Thus, as a consequence of the above the unsteady axial force may be obtained directly from the steady result by replacing a by the instantaneously effective angle of attack ав +ае c/2 V, that is,

ca{t) = rja Cn(r) tanj % Г)а Cna(M) (ae + . (8.173)

Notice that the noncirculatory loading does not appear in this expression. The unsteady pressure drag (if this is required) is obtained by resolving the total normal force (including noncirculatory terms) and axial force coefficients through the AoA a to obtain

Cd(t) = C„(t) sin a – Ca(t) cos a + C^. (8.174)

See also Leishman (1987b) for further discussion on the modeling of unsteady drag.