Category Fundamentals of Modern Unsteady Aerodynamics

In previous chapters, we have seen how the foundations of the aerodynamics were established and the developments were made in a little more than a century in this discipline in relation to the Aerospace Engineering applications. The progress is still continuing thanks to the advances made in wind tunnel and flight test mea­surements as well as the remarkable improvements achieved in computational means implemented in numerical simulations.

The knowledge provided by the classical aerodynamics is sufficient to deter­mine the aerodynamic performances of the high aspect ratio wings at low subsonic speeds and the low aspect ratio wings at supersonic speeds. On the other hand, as the speed or angle of attack increases and/or the aspect ratio decreases, we need modern concepts for aerodynamic analysis. The increase in cruise speeds causes unsteady fluid-structure interaction because of unavoidable elastic behavior of high aspect ratio wings, and it also causes the wing to reach critical Mach numbers because of compressibility effects at high subsonic speeds. The three dimensional aeroelastic analyses of such wings can be done with reasonable computational effort because of advances made in modern aerodynamics. In addition, the design of supercritical airfoils, which has the geometry to delay the critical Mach number, has made the high subsonic cruise speed possible for the civilian and military aircrafts with wings having high aspect ratio, low sweep, low induced drag and high L/D for almost more than a quarter of a century.

During the last quarter of the twentieth century, the numerical and experimental studies performed for predicting the extra lift caused by the strong suction of a separated flow from the sharp leading edge made the design and construction of the planes with delta wings which are highly maneuverable at high angles of attack possible. At higher angles of attack the wing rock may occur depending on the sweep angle. The recent studies emphasize the effect of the leading edge sharpness or roundness on the wing rock phenomenon.

One of the ultimate and ambitious aims of the aerospace industries is to design and construct very fast vehicles which are to take the vast distances between the major cities on earth in a few hours. The research and development branches of

U. Gulfat, Fundamentals of Modern Unsteady Aerodynamics, 307

DOI: 10.1007/978-3-642-14761-6_9, © Springer-Verlag Berlin Heidelberg 2010

major aerospace companies have been conducting research to design a fast aero­space plane which can travel a distance equivalent of the half of the earth cir­cumference in a couple of hours. All these designs are based on the sustainable hypersonic flight at upper levels of atmosphere. The concept of ‘wave-rider’ which was introduced more than half a century ago has become hot again because of its considerably high L/D values for sustainable hypersonic flight. The continuous hypersonic flight, on the other hand is possible only with powerful engines based on the supersonic combustion of fuels with very high heating capacities. The sustainable supersonic combustion, once thought to be out of question because of being unstable, first became possible under laboratory conditions since 1990s, and then were tested on small unmanned hypersonic vehicles for short durations after the introduction of flame control devices which can provide controls over time intervals less than a millisecond. However, so far most of the attempts made in sustainable hypersonic flight tests have failed. Since the costs of these tests are too high, to reduce the risk of failure it is necessary to go through intense and time consuming studies. In order to have most risk free tests, it is necessary to start with an adequate data base for the relevant flight conditions. This, naturally, requires large data base exchange among the countries which allocate substantial budgets for their aerospace development programs.

The advances made, during last two decades, in research and development indicate that the interest in aerodynamics is in two opposite directions. The first direction is the steady or unsteady flow analysis for very small sized objects, which may even operate indoors at low Reynolds number and at moderate to high angles of attack. The second one is the aerothermodynamics of the large sized aerospace vehicles which can cruise at very high altitudes with very high speeds.

The design and construction of unmanned light small sized air vehicles fall under the first direction mentioned above. Shown in Fig. 9.1 are comparative positions of the flying objects, ranging from very small to large, on a graph represented as the relation between the flight Reynolds number and the mass as modified from Mueller and DeLaurier (2003).

The small unmanned air vehicles are to fly and operate in Laminar flow regime as seen from Fig. 9.1. The flight of birds, however, occurs in laminar to turbulent transition. Both the small planes and the large jumbo jets flying in subsonic regimes function totally in turbulent flows. Shown in the left corner of Fig. 9.1, the flying insects, with their mass being less than a gram, generate lift and propulsion with flapping wings. In a hovering flight of insects, the free stream speed is zero; therefore, the maximum wing tip speed is taken as the characteristic speed for determining the Reynolds number. The flapping frequency of the wings is quite high for the considerably small wing span which makes the tip velocity still to yield a laminar flow. The flapping of wings for a hovering flight either occurs in a symmetrical forward and backward fashion with respect to a horizontal plane, or asymmetrical upstrokes and downstrokes with respect to an almost vertical plane (Wang 2005). In the first type of flapping the lifting force of the profile provides the hover, whereas in the second kind of flapping the hover is maintained with the drag generated by the profile. In addition, the experiments show that there is a

Fig. 9.1 Mass versus Reynolds numbers for the flying objects varying from very small to very large

sufficient lifting force generated by the wings flapping with amplitudes larger than their chords.

The sustainable forward flight with wing flapping is possible if the Reynolds number based on the free stream speed is larger than a critical value. Actually, for a thin airfoil at an effective angle of attack less than the dynamic stall angle, the product of the reduced frequency with the dimensionless plunge amplitude, kh, plays also an important role to get a propulsive force, Fig. 8.31, adapted from Gulcat (2009). The empirical criteria, in a laminar flow regime, to obtain a propulsive force with flapping becomes: log10(Re)*kh > 0.72, where Re is the Reynolds number based on the free stream speed. Below this value, negative propulsion is created. At higher angles of attack, where there is a strong leading edge vortex formation at very slow free stream velocities, the criteria to generate a propulsive force are based on the Reynolds number expressed, independently from the free stream speed, in terms of the fre­quency x and the airfoil chord c reads as: Re = xc2/v > 50 (Wang 2005). The first criterion is useful for cruising of the micro air vehicles, whereas the second criterion is helpful during the transition from hover to forward flight.

The purpose of defining a criterion for the sustainable flight conditions is to design, construct and operate small size air vehicles mainly capable of hover and/ or fly forward with flapping wings as is done in nature. In this respect, the principal aerodynamic challenge in Micro Air Vehicle design is recently stated, in the conclusions and recommendation section of NATO TR-AVT-101 publication, as the search for the greater robustness; namely, gust tolerance, maneuverability, and more predictable handling quantities such as capacity to hover or even perch rather than the pursuit of greater efficiency! (TR-AVT-101).

The second direction in aerodynamic research is the design of very large and very fast aerospace planes which operate in high altitudes. Obviously, because of compressibility, heating and the chemical decomposition of the air at very high speeds, the multidisciplinary concepts from thermodynamics and the chemistry must also be considered. Shown in Fig. 9.2 is the historical and comparative development of the air vehicle range, speed and the cruising altitudes adapted from Kuchemann (1978).

The air vehicles shown in Fig. 9.2 travel their indicated ranges R, which are expressed in terms of the earth’s diameter D, at about same time duration with cruising at given Mach numbers. At the upper right corner of Fig. 9.2, the ‘wave rider’ concept takes its position as the future aerospace plane to cruise at hyper­sonic speeds. The necessary steps to be taken with specific consideration to aerothermochemistry to develop such hypersonic planes are described in a paper by Tincher and Burnett (1994). In their work, they further study the capabilities of such a plane to maneuver with assistance of the gravity in the atmosphere of a planet while making interplanetary travel in the future. The research related to the hypersonic aerodynamics made in Europe and USA during last two decades is published under the title of ‘Sustainable Hypersonic Flight’ in AGARD-CP-600. The national and/or multinational aerospace programs mentioned in this confer­ence proceedings, however, are either continuing with delay or postponed or even canceled due to budgetary restrictions at the start of the new millennium. The more up to date version of Fig. 9.2 is given by Noor and Venneri (1997) in their book ‘Future Aeronautical and Space System’ published in AIAA series. In their work, the design and performance characteristics of single or multistage, faster than 12 Mach planes, which can orbit in the outside of our atmosphere, are provided. In this context, at Mach numbers less than 12, only the sub-orbital flights in the upper
atmosphere seem to be possible. In this context, the most recent review of the challenges and the critical issues concerning the reliability of a computational data and the limitations of the experimental data for hypersonic aerothermodynamics is provided in the extensive summary by Bertin and Cummings (2006).

Speculative and overall predictions based on the various sources about the future aerospace projects as well as on the different scientific endeavors are pro­vided by physicist Kaku (1998), who is a renowned Popular Science writer, in his recent book on visions (Kaku). As a futurologist, Kaku’s predictions on the future extends to the end of the twenty-first century wherein he sees the realization of projects related to even interstellar travel, which will increase our level of civi­lization to type I civilization according to the classification of civilizations defined by Nicolai Kardeshev.

At the beginning of this millennium an abominable act of terror committed with four hijacked midsize jetliners shocked the whole world and changed the direction of research and development in the western world drastically. This change, mainly concerning national security, affected the research areas in many disciplines as well as the direction of research in aerodynamics. The necessity of developing MAVs functioning outdoors as well as indoors have become significant in oper­ations related to the security of humankind for many years to come (TR-AVT – 101). In this context, the unsteady aerodynamic tools are not only applied to analyze propulsive forces for aerial vehicles but also for the possible presence of explosive trace detection at the human aerodynamic wake (Settles 2006) for avi­ation security applications in a nonintrusive and reliable manner.

The last but not the least of many applications of unsteady aerodynamics is the studies of power extraction from an aerohydrodynamically controlled oscillating – wing for the purpose of clean energy generation. The possibility of producing energy from sailing ships or from tethered power generators flying in the global jet streams may increase the available energy densities one order of magnitude higher than the current energy densities available with conventional windmilling tech­niques on the surface of earth or power from rivers and tides (Platzer and Sarigul – Klijn 2009).

[1]

s2 = t — 2(Mx + R)

ap

The finite wing flapping differs, especially for the low aspect ratio wings, from the 2-D oscillatory motions of airfoils because of the presence of the tip vortex which is likely to interact with the leading edge vortex of the wing. For the large aspect ratio wings, however, the strip theory, based on the quasi 2-D approach, can give the approximate values for the total lift and the propulsive force once the type of motion is described. During the flapping of the wing, since the heaving amplitude changes linearly along the span, the dynamic separation angle also changes from one strip to another as well. Therefore, one has to make sure that each strip does not experience the dynamic stall. If there is a dynamic separation present in any strip then the leading edge vortex must be checked for bursting so that it does not lose its suction force. In case of a lost of suction in any strip, the contribution coming from that strip to the lifting and propulsive force must be reduced from the total accordingly (DeLaurier). Based on their modified strip theory Mueller and DeLaurier give their predicted averaged total thrust coefficient as negative and it agrees well with experimental values for a specific wing at low reduced fre­quencies, i. e. k < 0.1, which indicates power reduction, i. e. windmilling. There is an over estimated positive thrust for k > 0.1, and the over estimation is as high as 10%, for the reduced frequency of k = 0.2. The theoretical and the measured lift coefficients remain almost constant with respect to reduced frequency, wherein the theory over estimates the lift coefficient about 15% compared to experimental values.

Further experimental studies were conducted to model the 3-D dynamic stall of low aspect ratio wings oscillating in pitch (Tang and Dowell 1995) and (Birch and Lee 2005). Tang and Dowell modeled a low aspect ratio wing with a NACA 0012 in periodic pitch, and they observed that results of their simple model showed qualitative similarities with the data of corresponding 2-D airfoil. Birch and Lee, on the other hand, investigated the effect of near tip vortex behind the pitching rectangular wing with NACA 0015 airfoil profile having aspect ratio of 2.5 at Re = 1.86 x 105 within the reduce frequency range of 0.09-0.18. Their experi­mental results indicate small hysteretic behavior during the upstroke and down – stroke motions for both the attached and the light stall oscillations. In case of deep stall oscillations, however, during upstroke the lift and the lift induced drag values increased with the airfoil incidence more than during downstroke for which the size of the tip vortex was larger compared to that of upstroke.

More detailed and extended wind tunnel as well as numerical study of oscil­lating finite wings was given by Spentzos et al. Five different wing geometry, varying from rectangular to highly tapered planforms with swept back tips, whose aspect ratios ranging from 3 to 10 and Reynolds numbers ranging from 1.3 x 104 to 6 x 106, are studied in dynamic stall conditions. The reduced frequencies of pitching oscillations range from 0.06 to 0.17. A light stall study of a rectangular wing with NACA 0015 section and with aspect ratio of 10 at Re = 2 x 106 and M = 0.3 indicates that hysteresis curves for the lift and the drag narrow down considerably from half span to the tip both for the experimental and the compu­tational results. At the tip region, however, there is a considerable positive shift between the experimental and the numerical results for the coefficients, which is attributed to the flexibility of the wing at the tip region (Spentzos et al. 2007).

The spanwise flexibility is also effective in thrust production of a pitching plunging finite wing (Zhu 2007). For a flexible wing, modeled as a thin foil in air, there is an initially sharp increase in thrust coefficient with increase in the stiffness of the foil, and it remains almost constant after dropping to a certain stiffness value. However, the efficiency shows a small increase with increasing stiffness. The increase in the average pitching angle decreases the amount of thrust but has an increasing effect on the efficiency of the foil. Nevertheless, for hydrofoils, where the calculations are performed for water, the thrust gradually increases with increasing stiffness, and the efficiency decreases slightly. The effect of average pitching angle is the same as it was for the case of air. The effect of spanwise flexibility on the thrust of a finite wing may change with the tip vortex and the leading edge vortex interaction which may enhance or weaken the leading edge suction force created by the foil. For more precise assessment, further investiga­tions for the wings with tip vortex reducing devices become necessary.

The frequency of the flapping plays additional essential role in finite wing flapping because of presence of the tip vortex. As the frequency of the flapping increases, the vortex generation frequency also increases during the creation of lift. The starting vortex, the tip vortices shed from the left and the right tips of the wing and the bound vortex on the wing itself altogether form a vortex ring during the downstroke. At the end of the downstroke, since there is no lift on the wing, the bond vortex becomes a stopping vortex as shown in Fig. 8.43.

The starting and stopping vortices are equal in magnitude but opposite in sens, and both are normal to the free stream direction. The size of the idealized vortex ring in Fig. 8.43 depends on the wing span and the frequency of the flapping.

Fig. 8.43 Starting and stop­ping vortex generated during the downstroke

For the case of high frequency flapping the starting vortex can not move down­stream away from the wing, therefore, it affects the lift unfavorably. On the other hand, once the wing is at its lowest position for upstroke, the effective angle of attack must create a lift generating vortex so that another starting vortex, which is in opposite sign with the stopping vortex, forms after a little lag. At the end of the upstroke, when the wing is its top position, a new stopping vortex, which is almost equal to the previously formed stopping vortex, and the new tip vortices are formed to make a new vortex ring. This way, once a cycle of motion is complete with downstroke and upstroke a ladder type wake, which consists of stopping and starting vortices, is generated as shown in Fig. 8.44a.

In the ladder type wake, which is produced by flapping finite rigid wing, the starting vortex having an opposite sign with the bound vortex causes delaying effects on the lift. In order to avoid this delay and not create vortices which are normal to flight direction, the length of span is reduced during upstroke with making use of spanwise flexibility. During downstroke the wing has a full span to give wider gap between the tip vortices whereas this gaps narrows down because of having smaller wing span during upstroke, which makes the strength of the tip vortex to remain the same. Hence, in an alternating manner, we observe one wide and one narrow tip vortex street, which in literature is called concertina type wake as shown in Fig. 8.44b, Lighthill (1990). In concertina type, unlike the ladder type, the periodic occurrence of wake vortices normal to the flight direction which plays a delaying effect in lift generation, disappears. Therefore, the spanwise flexibility, which generates concertina type wake pattern is preferable for man made flapping wings having high aerodynamic efficiencies similar to the efficiencies of the wings exist in nature.

The flexible wing flapping in oscillating airfoils provides aerodynamic benefits in terms of lift and thrust generation as well as providing inherently light structures (Heatcote and Gursul 2007). The real positive effect of the chordwise flexibility in forward flight is the prevention of the flow separation by means of reducing the effective angle of attack while changing the camber of the airfoil periodically. During plunge motion with large amplitudes, we can keep the effective angle of attack lower than the dynamic separation angle with flexible camber (Gulcat 2009a, b). If we assume a parabolic camber, whose amplitude changes periodically with za(x, t) = —a*cos mt x2/^2 for a thin airfoil as shown in Fig. 8.36, we can obtain the boundary layer edge velocity due to flexible camber as

If we give the plunging motion as h = — h*cos (ait), and the camber motion with 90° phase,

za(x, t) = —a(t)x2/b2, a(t) = a*cos(at + я/2), — b < x < b

this provides us with the effective angle of attack which is less than dynamic stall angle. Now, the effective angle of attack for the combined motion at the leading edge is determined as follows

ae = tan — [—(h + WaLE )/Um] (8.44)

where WaLE is the downwash at the leading edge caused by the time dependent camber change. Shown in Fig. 8.37 is the time variation of the propulsive force coefficient plots obtained including viscous effects for the flexible airfoil at Re = 104 and k = 1 for three different camber ratios: (a) a* =0.05, (b) 0.1, and

(c) 0.15. The corresponding average force coefficients are found as a) CT = — 0.3265, (b) —0.3316, (c) —0.3398, respectively. The ideal average force coef­ficients and the computed values are compared in Table 8.4 at associated effective angles of attack, all less than the corresponding dynamic stall angle, which is 29°.

According to Table 8.4, tripling the camber ratio from 5 to 15% results in only a 4% increase in the average force coefficient, that is, from —0.3265 to —0.3398. This shows that increasing the camber ratio does not produce a significant overall propulsive force increase for the case of a flexibly cambered airfoil undergoing plunge.

The viscous drag acting on the parabolically cambered thin airfoil is also obtained using the boundary layer equations. Equations 8.2 and 8.5 give the inertial values of the velocity vector v = ui + vj and vorticty a, which is neces­sarily used in skin friction calculations, in moving deforming coordinates attached to the body as a non inertial frame (Gulcat 2009a, b) as shown in Fig. 8.38. Let x-y be the rectangular coordinates attached to the body, and let П-g be the curvilinear local coordinates with surface fitted П coordinate’s tangent angle with x axis being ab and let g be parallel to z axis. At a given point (x, y) this yields x = П cos a1, and y = xsina1 + g, wherein the continuity and the vorticity transport respectively reads as

1 Ou Ou Ov

tana1 + = 0

cosa1 on og og

Ox u Ox Ox 1 o2x

+ +(v _ u tan a1) = 2.

ot cosa1 on o g Re o g2

The discretized form of Eqs. 8.45, 8.46 for boundary layer solutions can be written in a way similar to those given in Appendix 10 except for new coefficients resulting from the scale factors expressed in terms of the surface angle a1.

So far, we have seen the aerodynamic benefits of the chordwise flexibility for the case of the periodic camber variation normal to the chord direction. Next, we are going to analyze the flexibility effects as the maximum camber location varying along the chord. Let the camber geometry of the thin airfoil be as shown in

Fig. 8.38 Body fixed x-y coordinates, and body fitted П-g coordinates for a para­bolically cambered thin airfoil

Fig. 8.38, and let the maximum camber location vary periodically with time. According to Fig. 8.38, the camberline equation for a piecewise parabolic varia­tion with the maximum camber a located at p reads as

—a(x — p)2/(1 + p)2, x<p a(x — p)2/(1 — p)2; x

The time dependent downwash expression, w(x, t) = Oz/Ot + UOz/Ox with p = Op/Ot then becomes (Gulcat 2009a),

2ap (x — p)2/(1 + p)3 + 2a (x — p)(p — U )/(1 + p)2, x<p 2ap (x — p)2/(1 — p)3 + 2a (x — p)(p> — U)/(1 — p)2, x > p

(8.48)

The full unsteady lift coefficient can be calculated for a simple harmonic motion using Eq. 3.32a. However, even if we assume that the periodic movement for the maximum camber location is simple harmonic, according to Eqs. 8.47 and 8.48, both the camber motion and the associated downwash are periodic but they are no longer simple harmonic. Therefore, we have to be cautious while using the formulae derived for unsteady force and moment coefficients. Nevertheless, for oscillations with small frequencies as a first approximation we can use the concept of steady aerodynamics, i. e. p = 0, the piecewise integration of Eq. 8.48 with Eq. 3.31a from —1 to p, and p to 1 gives the sectional lift coefficient as

For the maximum camber location at the midchord, i. e. p = 0, equation gives ci = 2ap as expected. The boundary layer edge velocity for the quasi steady case reads as

Є = 1 ± -= 1 ±— —— І8ІП(abs(p—x) 1 P( 1 — x2 ) ( 1 —p2 ) +1 — xp) /

U U p

[(1 — p)(1 + p)]2+^/y—x [(2(1+x)—2p)(p2+1)p+8(p2 — p(1+x))sin—(p) —SpP1 — p2 /[(1 —p)(1 +p)]2.

(8.50)

Here, + is used for upper and — is used for the lower surfaces of the airfoil. As expected, for p = 0 which means that the maximum camber at the mid-chord Eq. 8.50 gives

– = 2a ———— (1 + x)=2a VT~—x2". (8.51)

U 1 + x

The steady sectional moment and lift coefficients obtained for an airfoil having 2% camber with its maximum camber location at p where -0.5 < p < 0.5 are given in Fig. 8.39. As observed in Fig. 8.39, the moment coefficient becomes positive for the p values which are of the mid-chord where lift coefficient increases significantly.

Shown in Fig. 8.40 is the steady surface velocity perturbation change with the location of the maximum camber. As expected, the peak value of the perturbation moves toward the mid-chord as the position of the maximum camber point moves the same way. Also shown in Fig. 8.40 is the surface velocity perturbation for a cor­rugated airfoil, bilinear in nature, with maximum camber location at quarter chord.

For non-negligible frequency values we have to consider p = 0, therefore, the downwash expression, w = w(t, x) must include the relevant terms known as quasi-steady aerodynamics, of expression Eq. 3.32a. The sectional lift coefficient then reads as

cqs = ai (sin-(p)/2-pJ 1-p2/2-(p2+2)sJ 1 – p2/3 + n/4)+bi (sin- (p)/

2-pJ 1 – p2/2 -J 1 – p2 + p/4) + C1 (sin – (p)-s/1 – p2 + p/2) +a2(-sin-(p)/2 + p/1 – p2/2+ (p2+2)sJ 1 – p2/3 + n/4) + b2(-sin-(p)/2 +p/1 – p2/2 + J 1 – p2 + p/4)+c2(-sin – (p) + /1 – p2 + p/2)

(8.52)

wherein,

a1 = 2ap/(1 + p)3, b1 =-2a1p + a1(1 + p)-2aU/(1 + p)2,

C1 = a1p2 – a1 (1 + p)p + 2aU/(1 + p)2 a2 = -2ap/(1 – p)3,

b2 = -2a2p + a2(1 – p)-2aU/(1 – p)2, C2 = – a2p2 – a2(1 – p)p + 2aU/(1 – p)2.

Fig. 8.40 Surface velocity perturbation variation with maximum camber location

The edge velocity for the quasi steady aerodynamics reads as follows

Ue U

j л/1 — x2ln 1 / (p — x)y/(1 — x2)(1 — p2 + 1 — px

— [c1 + b1 (1 + x)+A1 (1 + 2×2) /2] (sin —(p)+n/2) + [c1 + b1 (1 + x)+A1 (1 + 2×2) /2] (sin—(p)— p/2) + [(fl1 — a2)(p/2 — 2x)— b1 + b2W 1 — p2

(8.53)

The steady and the quasi steady aerodynamic approaches do not consider the effect of the wake as phase lag between the motion and the aerodynamic response such as lift or moment, and the reduction in their amplitudes. As we know, the measure of this lag and the amplitude reduction is the Theodorsen function C(k) = F(k) + iG(k). The amplitude of the lift coefficient for the quasi unsteady aerodynamics according to Eq. 3.32a reads as

cqu = C(k)cqs. (8.54)

The apparent mass term plays no role in quasi unsteady aerodynamics to give a simple relation between the vortex sheet strength and the lifting pressure, i. e. Ca = cpa/2. The boundary layer edge velocity then is found from the perturbation velocity: {u = ya/2}. The leading edge suction velocity P is given as

P = Jl^—A їа ‘ + ^

In expanded form it reads:

| — (c1 + 1.5a1 )(sin p + p/2) + (c2 + 1.5a2)(sin p — p/2) (p/2 + 2)a1 — b1 — (p/2 + 2)a2 — b2]/1 — p21

which is to be used in Eq. 8.27 to calculate the suction force. Knowing P from Eq. 8.55, the quasi unsteady lift from Eq. 8.54, and the equivalent angle of attack from quasi-steady lift, i. e. a = cl/2p, we can obtain the propulsive force S from Eq. 8.27.

The effect of the unsteady motion of the camber location is studied under various conditions for the maximum camber location changing with p = — 0.25[1 – cos (ks)], where s = Ut/b is the reduced time. Shown in Fig. 8.41 are the typical lift and thrust variation plots for the cambered thin airfoil having chordwise flexibility with maximum camber of 3% and reduced frequency of k = 0.2. The quasi unsteady lift and thrust coefficients shown with continuous lines indicates the expected phase lag between the motion and the aerodynamic response. Since the reduced frequency k = 0.2 is small, the differences among the steady, quasi steady and the quasi unsteady lift and thrust coefficients are not too large. According to Fig. 8.41, the maximum lift and the zero thrust are obtained for p = 0 for which the maximum camber is at the midchord, and the minimum lift and the maximum thrust are achieved when the maximum is at quarter chord.

The averaged suction force coefficients obtained by time integration of the curves over a period given in Fig. 8.41 are represented in Table 8.5 for (a) steady, st, (b) quasi steady, qs, and (c) quasi unsteady cases, qu. According to Table 8.5, the force coefficient becomes smaller for quasi-unsteady treatment with increasing reduced frequency.

For a flat plate at Re = 10,000 the drag coefficient according to Blasius is cd = 0.0266. The boundary layer solution obtained with the procedure as

described in Appendix 10, and based on the edge velocity given by Eq. 8.53, gives the viscous drag opposing to the motion as 0.0286 for k = 0.2 and 0.0266 for k = 0.8. This shows that the smallest propulsive force coefficient 0.0341, obtained with quasi unsteady approach for k = 0.8, for an airfoil morphing with a fixed camber ratio of 3%, easily overcomes the viscous drag produced by the chordwise flexible airfoil.

The chordwise change in the camber is considered simple harmonic. However, the associated downwash w given by Eq. 8.48 is no longer simple harmonic, especially for motions having high frequencies. Shown in Fig. 8.42 is the quasi steady lift, Eq. 8.52, change with time and the quasi unsteady lift obtained with the FFT applied to the equivalent motion whose angle of attack determined via Eq. 8.49 as an arbitrary motion. Comparison of Figs. 8.41 and 8.42 shows the effect of the reduced frequency, which is low for the small values of k, on the lift coefficient amplitude of the chordwise flexible motion, whereas the time averaged lift coefficient is almost the same for quasi steady and the quasi unsteady approaches as seen in Fig. 8.42.

The full unsteady approach includes the apparent mass term given by the second term of the right hand side of Eq. 3.27. The apparent mass term contributes to lift but makes zero contribution to leading edge suction term.

In this section we have analyzed the active chordwise flexibility of a thin airfoil. There are experimental, in water tunnels (Heatcote and Gursul 2007), as well as numerical studies based on fluid-structure interaction (Zhu 2007) concerning the passive flexibility with known or assumed elastic behavior of the thin hydrofoil flapping in water. The experimental and the numerical results agree well for the deformation of a thin and a thick flexible steel plate undergoing periodic heaving motion. The results obtained for a pitching plunging elastic airfoil by Zhu indicate that with increasing stiffness the thrust coefficient increases while the efficiency decreases. The effect of the maximum angle of attack is, however, opposite i. e. the

efficiency increases and the thrust coefficient decreases as the maximum effective angle of attack increases. The behavior of the steel plate in air as inertia driven deformation is somewhat similar at least qualitatively. However, for low stiffness values both the thrust and the efficiency are very small. Furthermore, the thrust becomes negative, which implies drag, for even lower values of stiffness.

In recent years, among the subjects of unsteady aerodynamics the flapping wing theory, which is based on the Knoller-Betz effect, has been the most popular one because of ever increasing demands in designing and manufacturing for micro aerial vehicles, MAVs (Platzer et al. 2008; Mueller and DeLaurier 2003).

In order to have sustainable flight with flapping wings, it is necessary to create a sufficient propulsive force to overcome the drag force as well as a sufficient lifting force. In finding the propulsive force we have to evaluate the leading edge suction force created in chordwise direction with pitching-plunging motion of the profile. If we model the profile as a flat plate undergoing unsteady motion, we can obtain the change of the suction force and the lifting force by time using the vortex sheet strength obtained via potential theory (Garrick 1936, von Karman and Burgers 1935). For the sake of simplicity, let us first analyze the plunging motion of the flat plate undergoing a simple harmonic motion given by h = hel0}t, where h is the amplitude of the motion. In terms of reduced frequency k, the Theodorsen function C(k) = F(k) + lG(k), and the non dimensional amplitude h* = h/b the sectional lift coefficient reads as

hi = -2nkh*C(k) + nk2h*. (8.26)

The lifting pressure distribution which provides this lift coefficient also creates a leading edge suction force in the flight direction. The relation between this suction force S and the singular value of the vortex sheet strength at x = — 1 reads as у = 2P/px* + 1, where P = p2C(k)h,

S =-(npP2 + a L). (8.27)

Here, a is the angle of attack, and L is the associated lift if there is also pitching. The derivation of Eq. 8.27 is given in Appendix 9. The minus sign in front of the suction force indicates that it is in opposite direction with the free stream, which means it provides a force in the direction of flight at pure plunge, and for pure pitching it may give negative propulsion depending on the phase lag between the angle of attack and the associated lift.

As an example using Eqs. 3.25-3.26, we can obtain the sectional lift and propulsive force coefficients for a flat plate in vertical unsteady motion given by h = —0.2 cos xt and the reduced frequency of k = 1.5. The real part of the sectional lift coefficient is created by the real part of h(t) which corresponds to cos xt. Therefore, with small manipulations we obtain for the lift

ci = -2pk[(G(k) + k/2)cos(ks) + F (k)sin(ks)]/** (8.28)

and for the suction

cs = —2nk2 [G(k)cos(ks) + F (k)sin(ks)]2 й*2. (8.29)

Here, s = Ut/b shows the reduced time. Shown in Fig. 8.24 is the time variation of the motion of the plate, sectional lift and suction force coefficients with respect to reduced time. During the simple harmonic motion of the plate, since the angle of attack is zero the sectional lift coefficient changes periodically with the amplitude of 1.7 and with the frequency of the motion but with a phase lag. When the profile is at its lowest position, the lift coefficient is negative, and during the early times of upstroke it decreases to its minimum -1.7. While it is still in upstroke motion, the cl value increases gradually to become positive as the profile reaches the highest position. During early stages of down stroke the lift coefficient starts to increase to reach its maximum value of 1.7, and then its value decreases to become negative as one cycle of motion is completed. In other words, as the bound vortex Ua on the plate changes in proportion with the lift, because of the unsteady Kutta condition there is a continuous shedding of vortices with the opposite sign to that of bound vortices into the wake.

During the down stroke of the airfoil, the clockwise rotating bound vortex grows in magnitude for a short time, and after its maximum value it gets smaller while a counter clockwise vortex is shed into the wake from the trailing edge. After the profile passes the midpoint location, the sign of the bound vortex changes to become a counterclockwise rotating vortex while a clockwise rotating vortex is shed into the wake. The schematic representation of the bound vortex formation and the vortex shedding into the wake is shown in Fig. 8.25. In Fig. 8.24, shown is the sectional suction force variation by time which indicates that the propulsive force coefficient remains 0-0.2 in magnitude while its frequency becomes the double of the frequency of the motion. The maximum values of the propulsion

Fig. 8.24 Lift ci and the suction force cs coefficient changes with the vertical motion h of the profile

x

occur as the profile passes through the midpoint during its down stroke, and the zero propulsion is observed twice right after the top and bottom points of the profile’s trajectory in one cycle. This shows us that the creation of the maximum suction force occurs with 90° phase difference with occurrence of maximum or minimum bound vortex. That is when the absolute value of the bound vortex is highest the profile produces zero suction force.

The shedding of vortices in alternating sign from the trailing edge to the wake as described above forms a vortex street. The vortex street in the wake of the oscillating flat plate as shown in Fig. 8.25 indicates that the vortex shed at the top position of the airfoil is in conterclockwise rotation, and the previous vortex shed at the bottom location is in clockwise direction. This means the vortex street has counterclockwise rotating vortices at the top row and clockwise rotating vortices at the bottom row. We note at this point that the vortex street forming at the wake of vertically oscillating flat plate is exactly opposite to the vortex street forming behind the stationary cylinder where the top row of vortices rotate in clockwise and the bottom row vortices rotate in counterclockwise direction. The vortex streets generated behind the circular cylinder and at the wake of the oscillating flat plate have been also observed experimentally (Freymuth 1988). It is a well known fact that the wake formed behind the cylinder creates a drag on the cylinder whereas the wake of the oscillating plate has a structure which is opposite in sign is naturally expected to give a negative drag i. e. propulsion!

Now, let us analyze the physics behind the creation of propulsive force by a vertically oscillating profile using the concept of the force acting on a vortex immersed in a free stream as shown in Fig. 8.26. During the down stroke a clockwise rotating bound vortex is experiencing a vertical velocity component equal to Uz = h for the cases (a) the approximate suction force of S * pUz Ta, and during the up stroke the counterclockwise rotating bound vortex is under the influence of vertical velocity which is in — z direction to create (b) S * pUz Ta which is the approximate suction force. Here, during (a) down stroke, and (b) up

Fig. 8.26 The generation of suction force S during a downstroke, b upstroke

stroke motions the vertical velocity component and the bound vortex change simultaneously so that the suction force S remains in the same direction as a

bound circulation Uz Ta remains the same, its magnitude changes by time as shown in Fig. 8.24.

The time and space variation of the wake vortex sheet strength can be computed in terms of the bound vortex using the potential theory. The relation between the vortex sheet strength yw and the bound vortex Ca can be established using Eq. 3.13 3.13 for a periodic motion of the profile given by za = h cos ks as follows

(8.30)

[(H(2) sinkx + H02) coskx)cosks + (H(2) coskx + H02) sinkx) sinks].

Now, with the aid of Eq. 8.30, we can show the spacewise variation of the wake vortex sheet strength at the top and bottom positions of the profile on Fig. 8.27. As shown in Fig. 8.27, at the bottom position of the profile the shed vortex is positive

i. e. in clockwise direction, and at the top position it is negative i. e. in counter­clockwise direction. The near wake region vortex signs are in accordance with the signs given in Figs. 8.25 and 8.26 which is indicated in the experimental results of Freymuth.

The propulsive efficiency of the flapping wing is another concern to the aero – dynamicist. In order to calculate the average propulsive efficiency in one cycle, we have to know the average energy which is necessary to maintain the propulsion and also the average work for the vertical periodic motion. The ratio of the average energy to average work gives as the propulsive efficiency. Accordingly, for a

periodic motion given by za = — h cos xt, with the aid of Eqs. 8.28 and 8.29 we obtain

/2p/x SUdt F2 + G2 /2p/x Lhdt ~ F

Shown in Fig. 8.28 is the variation of the propulsive efficiency with respect to the reduced frequency k.

The theoretical results obtained for the lift and the suction forces of a vertically oscillating thin airfoil at zero angle of attack are in agreement with the solutions obtained using Navier-Stokes equations for NACA 0012 airfoil in plunging motion (Tuncer and Platzer 2000). Solutions based on the potential flow assumptions and the Navier-Stokes solutions give similar results for the amplitude and the period of both the lift and the suction forces. Naturally, Navier-Stokes solutions also provide viscous and form drags. On the other hand, using the unsteady viscous-inviscid coupling concept and the velocity viscosity formulation the skin friction of the thin airfoil can be determined with numerical solution of the Eqs. 8.5-8.7 with the boundary layer edge velocity Ue = Ue(t) provided by the potential flow as described by Gulcat (Problem 8.28 and Appendix 10). As shown in Fig. 8.28, the difference between the theoretical and the numerical solutions is apparent for the values of propulsive efficiency. The efficiency obtained by the ideal solution is independent of the plunge amplitude, and becomes very high for low frequency oscillations and asymptotically reaches the value of 0.5 for very high frequencies. Obviously, viscous solutions yield lower values of efficiency, and they depend on the amplitude of plunge as shown in Fig. 8.28. The efficiencies obtained with viscous effects indicate that for high plunge amplitude the efficiency values show the tendency to follow the ideal curve. However, for the plunge amplitudes less then 0.4 the efficiencies become small with decreasing of fre­quencies as opposed to the ideal case, whereas the efficiency obtained with
boundary layer approach is in between the ideal and the Navier-Stokes result. According to Navier-Stokes solutions for the efficiency to be more than 0.5, the condition must be k < 0.6 and h > 0.4.

Previously, we have seen that the dynamic stall takes place at higher angles of attack than the occurrence of static stall depending on the reduced frequency values. The higher the reduced frequency, the more the difference between the static and dynamic stall angles. For a pitching airfoil, the difference between the static and dynamic stall angles Da in terms of the reduced frequency k is given empirically as follows (Prouty 1995)

aDY — aST = Da = yVk, у = 0.3 — 0.5 (8.32)

On the other hand, as seen on Fig. 8.6, NACA 0012 profile at the reduced frequency value of k = 0.15 can undergo pitching without flow separation up to 20° angle of attach. Above that, between 20° and 23.5° angle of attack, there is a leading edge separation which generates a vortex causing a high lift until the vortex is convected to wake from the trailing edge. Static wind tunnel experiments show that the flow over the profile separates at 13° angle of attack. The 7° dif­ference between the static and dynamic stall angles is slightly higher than the empirically estimated value obtained by Eq. 8.32 using the lower value of coef­ficient у. That is to say Eq. 8.32 gives a little bit conservative estimates for the dynamic stall angles of pitching airfoils. The effective angle of attack for the plunging airfoil in a free stream of U becomes zero at the top and bottom locations, and takes its maximum value at the center point. During down stroke the effective angle of attack gives positive lift, and during up stroke it provides negative lift. Now, we can calculate the relation between the effective angle of attack, plunge amplitude and the frequency for an airfoil undergoing vertical oscillations za(t) = —hcos(mt) in a free stream as follows. Since he vertical velocity of the airfoil then becomes Za = hmsin(mt), the effective angle of attack reads as

tanae = -—у = —k(h/b). (8.33)

According to Eq. 8.33, the effective angle of attack is proportional with the product of the free stream and the dimensionless plunge amplitude. The dynamic separation angle as given in Eq. 8.32 depends on only the reduced frequency. The airfoil pitching with reduced frequency of k = 1.5 has the dynamic separation angle with aDY = aST + Da = 13 + 21 = 34°. This means at the reduced k = 1.5 the profile can undergo plunge oscillation up to the non dimensional plunge amplitude h/b = 0.45 without experiencing flow separation if we consider the plunging with the effective angle of attack is equivalent to the pitching with the same angle of attack. This assumption lets us apply the potential flow theory for a wide range of plunge rates with boundary layer coupling to take the viscous effects into account.

Using Eqs. 8.32 and 8.33 we can find the maximum plunge amplitude in terms of the reduced frequency for a profile encountering no flow separation as given in

Fig. 8.29 Change of plunge amplitude with the reduced frequency without experienc­ing dynamic stall

) = 1 ±[F(k)sin(ks) + G(k)cos(ks)]kh*p(1 — x)/( 1 + x) . (8.34)

The addition of the suction force and the drag gives us the instantaneous pro­pulsive force coefficient as CF = cs + cd. The time average of CF over one period gives us the definition of the average propulsive force CT as follows:

T

cT = T j CFdt. (8.35)

0

Table 8.1 gives the averaged propulsive force coefficients obtained for various plunge amplitudes with the viscous-inviscid interaction, and compares with the results obtained with N-S solutions for NACA 0012 airfoil at Reynolds number of 105 and the reduced frequency of k = 0.4.

For viscous-inviscid interaction to be applicable at k = 0.4, Eq. 8.32 dictates that the effective angle of attack ae should be less than the dynamic stall angle of NACA 0012, i. e. ad = 12° + 0.3 (0.41/2) & 23°. For angles of attack larger than 23°, as seen from Table 8.1, the viscous inviscid interaction overestimates con­siderably the averaged propulsive force. The thickness effect is also important in prediction of drag on an airfoil. If the thickness correction (Van Dyke 1956) is

made for the NACA 0012 airfoil, the agreement between the viscous inviscid solution becomes very good for the low effective angles of attack.

We know now the capabilities and limitations of viscous-inviscid interaction approach for plunging thin airfoils. Therefore, we can perform parametric studies to predict the average propulsive force depending on the Reynolds number, plunge amplitude and the reduced frequency. The wind tunnel experiments indicate that to obtain a net propulsive force for a plunging airfoil the product of the reduced frequency and the dimensionless plunge amplitude must be higher than a critical value, i. e. kh > 0.2. where the Reynolds number is 17,000 (Platzer et al. 2008). The Reynolds number, however, is also an important parameter to obtain net propulsive force as shown in Fig. 8.30.

The variation of the average propulsive force coefficient, cT for different dimensionless plunge amplitudes h/b = 0.2, 0.4, and 0.6 is given in Fig. 8.30a-c respectively. Figure 8.30a indicates that, for h/b = 0.2 to generate a net propulsive force, the Reynolds number must be greater than 103 and the reduced frequency must be greater than 1.2. If the plunge amplitude is doubled, that is, for h/b = 0.4 according to Fig. 8.30b, for the reduced frequency values greater than 0.5, a net propulsive force is obtained even for a Reynolds number of 103. Moreover, increasing the amplitude to 0.6 gives a net propulsive force for a wide range of frequencies, that is, k > 0.3, and Re > 103 as shown in Fig. 8.30c. A close inspection of Fig. 8.30b, c indicates that when the amplitude is high, the increase in the Reynolds number from 104 to 105, that is, one order of magnitude increase has very little effect on the propulsive force coefficient. Figure 8.31 shows the Reynolds number dependence with kh variation of the net propulsion generation of a plunging thin airfoil. The region above the line indicates propulsion whereas below the line there is a power extraction area which is of importance to wind engineering when performed as pitching and plunging for significant power extraction for clean energy production (Kinsey and Dumas).

Finally, for the plunging airfoil we can give the propulsive efficiency values obtained with the viscous inviscid interaction. Table 8.2 shows the comparison of the efficiency values with the Navier-Stokes solutions of (Tuncer and Platzer 2000).

According to Table 8.2, there is an 8% difference for the efficiency with the viscous inviscid interaction and the full N-S solution at 80% plunge amplitude with respect to the chord. This discrepancy becomes 18% for 100% plunge amplitude because of having high effective angle of attack where N-S solution predicts weak separation at the trailing edge.

Fig. 8.31 Reynolds number and kh dependence of the propulsion and power extraction

So far, we have seen in a detail, lift and propulsive force variations of a plunging airfoil as a one degree of freedom problem. As a result, at zero angle of attack, the lift created is positive during down stroke and negative during up stroke to give zero average value, and the propulsive force is generated for a certain range of kh values and Reynolds numbers if we take the viscous effects into consider­ation. In order to obtain positive lift throughout the flapping motion two degrees of freedom, i. e. pitching and plunging becomes necessary for the airfoil. We can impose a pitching plunging motion on the airfoil for which the lift is always positive because of effective angle of attack if we describe the pitching with a, the plunging with h and the phase difference between the two with y as follows:

h = —icosmt a = a0 + hcos(rot + y)

The unsteady motion of the airfoil given by Eq. 8.36a, b the effective angle of attack at the leading edge of the airfoil reads as

where d is the distance between the leading edge and the pitch axis. If we consider the pitching over a constant angle of attack, during up stroke if we let the angle of attack increase and during down stroke let the angle of attack decrease then we can have an effective angle of attack always positive during the forward flight given by Eq. 8.37, which yields positive lift throughout the pitch and plunge. Now, we can illustrate the whole motion on a simple figure as the superpositioning of Eq. 8.36a, b, as depicted on Fig. 8.32a, b during (a) down stroke, and (b) up stroke.

According to Fig. 8.32, during (a) down stroke, and (b) up stroke, the effective angle of attack shows very little change. If we can keep the effective angle of attack given by Eq. 8.37 lower than the dynamic separation angle, we can use the viscous-inviscid interaction to predict the propulsive and the lifting forces of a pitching plunging airfoil. Problem 8.29. Detailed numerical studies of a pitching plunging airfoil were given in late 1990s as Euler and Navier-Stokes solutions at Re = 105 (Isogai et al. 1999), and comparison is made with the Lighthill’s

potential solution. Isogai et al. studied the motion of NACA 0012 airfoil in dimensionless plunge amplitude of 1.0, angle of attack amplitude of 20°, pitch axis location as the midchord, and the phase angle as 90° to calculate the propulsive force coefficient and the efficiency in terms of the reduced frequency k. Naturally, the highest efficiency is obtained with potential theory, and the Euler and N-S solutions yield less values of efficiency respectively at k values ranging 0.5-1.0. As it happens for the case of pure plunge the efficiency decreases with increasing k for the pitch-plunge case. As k changes in 0.5-1.0, the efficiency of the potential flow ranges in 0.85-0.75, Euler solution gives 0.8-0.6, and N-S yields 0.7-0.55, respectively. For the Navier-Stokes solutions, there is no significant efficiency variations for laminar and the turbulent cases. At the same range of reduced frequency, the propulsive force coefficients vary between 0.4-0.6 for the potential solution, 0.35-0.75 for Euler, and 0.3-0.6 for N-S solutions. These results indicate that the propulsive force coefficient increases with increasing reduced frequency. The Navier-Stokes solutions performed by Tuncer and Platzer (1996) under similar flow conditions agree well with the work of Isogai et al. However, at phase difference of 30°, there is a discrepancy between two approaches as far as the leading edge separation of the solution given by the latter is concerned.

As indicated with Eq. 8.27, pure plunging always creates a leading edge suction which yields a propulsive force. However, it is not so for the pure pitching motion of an airfoil because of the phase lag between the angle of attack a and the lifting force L. This phase lag may yield negative average propulsion, i. e. drag even with potential flow analysis, depending on the position of the pitch axis a for all ranges of reduced frequency. Let us consider the pure pitching motion with a = acos(rnf + u), wherein only the second term of the right hand side of Eq. 8.36a, b is considered. The averaged propulsive force from Eq. 8.27 and 8.31 with the aid of (Garrick 1936), and with small correction, reads as

4; (k; a) = p(F2 + G2)

Shown on Fig. 8.33 are the curves for the averaged propulsive force coefficients plotted against the inverse of the reduced frequency.

According to Fig. 8.33, by definition, negative values of averaged propulsive force indicate propulsion whereas the positive values mean the fluid extracts power from the pitching airfoil. For the pitch axis at three quarter chord, i. e. a = 1/2, at all values of reduced frequency there is not any propulsion predicted. At large values of k the pitching about the leading edge a = — 1, the quarter chord point a = —1/2, the trailing edge a = 1, and the mid chord a = 0, we observe that it is not possible to generate propulsion. However, for small values of k, i. e. k < 1, we see that except for a = 1/2, generation of propulsive force is possible. Therefore, according to the ideal theory, if we want to have contribution to the propulsive force from the pitching, it is necessary to choose a proper pitch axis as well as the

reduced frequency range for o pitching plunging airfoil. This adverse effect of pitch axis location on the propulsive force naturally alters the propulsive effi­ciency. The ideal efficiency formula for the pitching plunging airfoil, Eq. 8.36a, b, with the phase difference of u can be obtained as

_ /0 ^ SUdt ^ ah*2 + (a2 + b2)a + 2(a4 + b4)h*a 394

g /2p/x (Lh + Ma)dt c1h*2 + c2 a2 + 2c4h* a ‘

where:

a4 — a1 [1/ksin(u) + (0.5 — a) cos(u)] — 0.5Fcos(u) — 0.5Gsin(u),
b2 — —0.5a — F/k2 + (.5 — a)G/k, b4 — 0.5(.5 + G/k)cos(u) — F/ksin(u),
C1 — F, C2 — 0.5(0.5 — a) — (a + 0.5)[F (0.5 — a) + G/k],

C4 — 0.5(0.5 — 2aF + G/k)cos(u) + 0.5(F/k — G)sin(u).

Knowing that the pitch may hamper the propulsive efficiency we have to choose the pitch axis with caution as well as the phase between the pitch and plunge. Equation 8.39 gives the ideal propulsive efficiency g = 0.87 for a flat plate pitching about mid chord with k = 1, h* = 1.5, a = 15° and u = 75°, whereas g = 0.54 is computed with N-S solution for NACA 0012 airfoil at Re = 104 and with the same flow parameters (Tuncer and Platzer 2000). There exist further studies, based on the N-S solutions, to optimize the efficiency and/or thrust in terms of plunge magnitude, pitch magnitude and the phase lag (Tuncer and Kaya). Tables 8.3 and 8.4 show the comparison of the optimized propulsive efficiency computed using N-S solutions for NACA 0012 airfoil at Re = 104 with the ideal efficiency calculated using Eq. 8.39 for an airfoil pitching about its midchord.

According to Tables 8.3 there is a 20% difference in the ideal efficiency and the efficiency evaluated with N-S solutions, and the efficiency increases with increasing pitch amplitude. Furthermore, solving for maximum efficiency may not yield a good thrust coefficient as well as searching for maximum thrust may not produce very high efficiency.

Now we are ready to give examples to evaluate the effective angle of attack of a pitching plunging airfoil for various h, a and k values for which dynamic sepa­ration angles are larger than the effective angle of attack.

Example 8.5 Assume an airfoil pitching about its leading edge and plunging with k = 0.35 as follows

h = 1.1cos(xt) a = 10° + 10°cos(x + n/2)

Solution: Since the reduced frequency is given we describe the motion in reduced time with following equations:

h = 1.1cos(ks) a = 10° + 10°cos(ks + n/2)

Taking d = 0 for Eq. 8.37 gives the expression for the effective angle of attack ae = ae(s), whose plot for a period of motion is given as follows:

According to Fig. 8.34, the effective angle of attack remains less than 23° which is under the dynamic separation angle given for NACA 0012 profile with Eq. 8.32. That means the profile can undergo high amplitude pitch and plunge without encountering separation. During down stroke, the angle of attack gets smaller but the relative air velocity in vertical direction causes increase in the effective angle of attack. During up stroke, however, the increase in angle of attack makes the effect of the negative vertical air velocity vanish. As a result of this pitch and plunge it becomes possible to have an unseparated flow throughout the motion because of having the effective angle of attack under 20°. At the same time, the angle of attack and the effective angle of attack remains positive to yield a positive lift. It is necessary to make a note here that according to Fig. 8.33 the propulsion due to pitch is also favorable because of pitch axis location and the k being 0.35.

Example 8.6 The NACA 0012 airfoil is pitching and plunging with a reduced frequency of 1 as given below

h = 0.65cos(xt) a = 20 – b 20 cos(xt – b p/2).

Show that the effective angle of attack remains under the dynamic separation angle of attack.

Solution: The dynamic separation angle of attack is found as

acr = 13° + 0.3л/k * 180/p = 30°. In terms of reduced time s it reads as

h = 0.65cos(s) a = 20° + 20° cos (s + p/2).

The superposition of pitch and plunge gives us the effective angle of attack less than 30° as shown in Fig. 8.35. Since the effective angle of attack remains above 10°, the instantaneous lift is always positive and relatively high. As seen from Fig. 8.35 during the flapping motion relative to free stream the angle of attack changes between 0° and 40°.

The results of Examples 8.5, 8.6 indicate that: (a) for low reduced frequencies, i. e. k 1, pitching with small angles of attack and plunging with high amplitudes and with 90° phase angle we obtain effective angles attack less than the dynamic separation angle, (b) for k > 1 with small plunge amplitudes and large angles of attack, flapping without exceeding the dynamic separation angle is possible.

So far we have studied the pitch plunge motion of an airfoil prescribed as simple harmonic motion. However, a nonsinusoidal motion of the flapping airfoil is also observed to yield sufficient propulsive force through path optimization (Kaya and Tuncer 2007). In their study Kaya and Tuncer used B splines for the periodic flapping motion. They showed that thrust generation may significantly be increased, compared to the sinusoidal flapping, with the characteristics of the path for optimum

thrust generation staying at about constant angle of attack at most of the upstroke and downstroke, while pitching is happening at extremum points of plunge.

We know now that in order to create a propulsive force we need to create a reverse Karman vortex street at the wake of the oscillating airfoil. The creation of the reverse Karman street is possible either with attached flow or with flows creating strong leading edge vortices which in turn generate appreciable leading edge suction. If the leading edge vortex formed, because of angle of attack exceeding the dynamic separation angle, does not burst at the trailing edge, it will create considerable suction at the upper surface which will help for propulsion and lift as well. As seen in Figs. 8.6 and 8.7, the N-S solutions which are in agreement with experiments, show increase in lift although the dynamic separation angle is exceeded by 30-40. Further increase in the angle of attack creates bursting of the vortex at the trailing edge to cause lift lost. However, if the reduced frequency is increased above 0.15 it is possible to go to higher angles of attack without causing vortex burst at the trailing edge (Isogai et al. 1999). At high Reynolds numbers, laminar or turbulent, it is possible to create a propulsive force without resorting to high angles of attack. On the other hand, at low Reynolds numbers, i. e. Re < 1,000, the pitching motion may provide propulsion at low frequencies if the angle of attack exceeds 200. For this case maximum thrust is achieved in 450-600 angle of attack range (Wang 2000).

The last aspect of the pitching plunging airfoil to be briefly mentioned here is the power extraction from the oscillating airfoil (Kinsey and Dumas 2008). This time rather than having propulsion with the unsteady motion which is provided by the energy of the fluid, the energy will be given to the fluid by the motion of the airfoil to generate power which is useful in harvesting wind energy. The pitch plunge motion here is conventionally defined with a(t) = a sin(xt), and h = asin(xt + u) with the approximate definition of the feathering parameter (Anderson et al. 1998; Kinsey and Dumas 2008)

a

tan-(xh/U)

which is approximately associated with propulsion for v < 1, whereas v > 1 corresponds to power extraction, and naturally, v = 1 yields neutral motion called feathering for which neither propulsion nor power production exist. If the average power extraction coefficient over a cycle due to plunge and pitch combined is denoted with Cp then the power extraction efficiency reads as

P – b

g = it = Cpt (8.41)

Pd h

where P is the total power produced and Pd is the total power of the oncoming flow passing through the swept area during plunge. The power extraction efficiency is theoretically limited by 59% from a steady inviscid stream tube, whereas Kinsey and Dumas report about 33% efficiency and almost 2.82 average total force coefficient for NACA 0015 airfoil pitching about its 1/3 chord with b/h = 2, a = 76.33° and k = 0.56 at Reynolds number of 1100.

Wings with low aspect ratio, high sweep and sharp leading edges at high angles of attack undergo a self induced unsteady rolling periodic motion called ‘wing rock’. In water and wind tunnels, wings having a single degree of freedom, roll only, experiences wing rock beyond a critical angle of attack. The symmetrically formed leading edge vortices, beyond the critical angle of attack can no longer remain symmetric; therefore, their strength changes to create a moment about the axis of the wing. This moment, initially being small, causes wing to roll in one direction. Meanwhile, the vortex on the other side of the wing gets stronger and opposes the roll so that the motion reverses itself. Therefore, a self induced periodic motion is generated. In flight conditions, this rolling takes place together with the side sway and the plunging degrees of freedom. In Fig. 8.14, shown is the three degrees of freedom motion consists of (a) roll, (b) side sway, and (c) vertical displacement.

According to Fig. 8.14, the flight direction is in out of y-z plane, and the aircraft rolls about its axis while it moves sideways and descends with high angle of attack. The analysis of this three degrees of freedom motion is possible by evaluating the lifting force, sideways force, and roll moment acting on the body at every instant of the flight in interactive manner. In recent years, the three degrees of freedom problem based on the numerical solution of Euler equations to predict the aero­dynamic forces and moment acting on an aircraft in wing rock appeared in liter­ature (Saad and Liebst 2003). On the other hand, the numerical solution of Navier – Stokes equations for delta wings in free or forced rolling oscillations first appeared in the mid 1990s (Chaderjian 1994; Chaderjian and Schiff 1996).

Results of years of experimental as well as numerical studies on unsteady aerodynamics are summarized in Fig. 8.15. On the left side of the graph, where all low aspect ratio wing data was presented, the wing rock occurs above a certain angle of attack for the wings having leading edge sweep more than 74° (Ericsson 1984). If the sweep angle is less than 74°, instead of formation of leading edge vortices we observe their bursting. The bursting of a leading edge vortex causes

Fig. 8.14 Wing rock with three degrees of freedom: roll, side sway and plunging

> y

1.5 і і

k=m b/U

suction loss on one side of the wing which in turn creates a dynamic instability which is called roll divergence (Ericsson 1984). Starting of roll divergence, however, spoils the periodic rocking motion and causes wing to spin about its own axis. On the right side of the graph, the large aspect ratio effects are visible in terms static and dynamic stall limits.

The numerical studies on wing rock first became possible by modeling the leading edge vortex with unsteady vortex lattice methods (Konstadinopoulos et al. 1985). In that work, the equation of roll motion was based on the conservation of roll moment with roll angle and its time rates. The roll moment equation in terms of the non dimensional moment coefficient Ct as follows

Iф = 1/2pcAU2Ct(ф, ф, a) — і/. (8-17)

Here, I is the roll moment of inertia of the wing, ф is the roll angle, A wing surface area, c root chord and і is the bearing resistance to roll. Rearranging the coefficients in Eq. 8.17, and indicating the reduced time by s = 8Ut/c the non dimensional form of Eq. 8.17 reads as

/"(s) = CiCt(/, Ф, a) —С2Ф’ (8.18)

Here, the non dimensional coefficient C = pc3S/(128I) and C2 = ic/(8I). The roll motion is started with the non zero angle ф0 and zero angular velocity. With the sweep angle of 80° and root chord of 42.9 cm the delta wing is set to roll motion at various angles of attack to result in: (a) damped rolling for the angle attack less than 15°, (b) unstable periodic roll motion for the angle of attack more than 20°.

The simple analytical model construction of roll moment helps to analyze the wing rock phenomenon. Now, we can write a general expression for the non dimensional roll moment coefficient in terms of roll angle ф and the time rate of change of that angle ф0 as follows

C — & ф b a2ф b аз ф – Ь a4 ф ф – Ь a 5 ф ф – Ь a6 ф b a7 ф – Ь a 8 ф ф b a9 ф ф

Here, non dimensional coefficients a are computed by least square method. The first six terms on the right hand side of the Eq. 8.19 contribute significantly to the value C1, and the rest of the terms are insignificant. In addition, the terms with odd powers of ф constitute the restoring force, and the odd powers of ф are responsible for damping of the roll motion. Accordingly, we can write down the force coef­ficient CR which is responsible for the restoring force, the coefficient CD which is for the damping as follows

Cr — а ф b a3ф ~b a5ф ф (8.20a)

and,

Cd — a2ф b a4ф ф b a6ф • (8.20b)

Example 3 At 250 angle of attack, a wing with 80° angle of sweep rocks with period of 0.39 s. and amplitude of 32°. Obtain the graph of Cr versus ф and Cd versus ф, and comment on them.

Data: a1 = -0.0572, a2 = 0.1362, a3 = 0.0514, a4 = -1.403, a5 = -1.943, a6 = 0.075.

Solution:

1. The data is used to express the roll angle in radians ф = -32 sin (2pt/0.39)p/ 180 and writing non dimensional ф, the restoring force coefficient reads as

Cr — —0.0572ф + 0.0514ф3 – 1.943<ф2ф

whose graph is shown Fig. 8.16 which gives the force coefficient in opposite phase with roll angle. These two being in opposite phase make the motion

Fig. 8.16 Change in restor ing force for the wing rock 10Cr and ф by time

continue in a stable manner. The roll angle changes sinusoidally, however, because expression Eq. 8.20a being non linear, CR is periodic but no longer simple harmonic, as seen in Fig. 8.16, especially at the flat peaks of CR curve.

2. The aerodynamic damping coefficient is obtained by subtracting the bearing resistance from Eq. 8.20b as CD = 0.1362 / — 1.403/2/ + 0.075/3 — 0.004/. Shown in Eq. 8.18 is the aerodynamic damping coefficient CD and roll rate / with time. According to Fig. 8.17 the period of damping coefficient is the half of the period of roll rate. (The equations are used with permission of the ‘‘American Institute of Aeronautics and Astronautics’’).

According to Figs. 8.16 and 8.17, when the roll angle is approximately zero and the roll rate is near maximum, the damping moment and the roll rate have the same sign, and when the roll angle is maximum and the roll rate is zero, the damping moment and the roll rate have opposite signs. For this reason when the roll rate is maximum, since it has the same sign with the damping moment, there is a loss in damping which means there is a positive feeding of the motion. That is how the wing rock is sustained.

Example 4 Plot the hysteresis curve for the roll moments roll angle for the delta wing given in Example 3. Indicate the intervals on the hysteresis curve where the motion is damped and where it is fed. Compare the new plot with the comment made on the previous graph.

Solution: The total rolling moment coefficient is

CT = CR + CD

whose graph with respect to roll angle / is plotted in Fig. 8.18. In that figure when the curve follows the clockwise pattern, there is a negative damping, and the counterclockwise pattern there is a damping. Accordingly, as far as the intervals are concerned, in -32 < / <-18 and 18 < / < 32 there is

damping and in -18 < ф < 18 feeding occurs. A similar conclusion is made at high roll angle where there is a negative damping observed.

Examples 3 and 4 provided us with detailed information about the rolling moment change with roll angle of a wing in a wing rock as a single degree of freedom problem. During the rolling motion of a wing, except at zero yaw angle, while the effective angle of attack changes, the free stream direction also changes with an amount b as the yaw angle as indicated with 3-D representation on Fig. 8.19.

Now, let us express the effective angle of attack ae, and the effective yaw angle be in terms of roll angle ф = Аф sin xt. Here, Аф is the amplitude of roll angle.

For ф = 0 yaw angle we take the angle of attack as a0. During the rolling motion we consider only the rotational degree of freedom around the root chord of the delta wing. However, Eqs. 8.21 and 8.22 indicate that as the effective angle of attack deceases, the emerging sideways flow causes the flow symmetry to be spoiled. For this reason, the normal force acting on the delta wing changes during rolling and also because of spoiling of symmetry. Shown in Fig. 8.20 is the

variation of the effective angle of attack and the yaw angle with the change of roll angle as given in Examples 3 and 4 for a0 = 250 and Аф = 32°. Accordingly, the effective yaw angle changes between -12.5° and 12.5°, while effective angle of attack varies between 22° and 25°. Since the effective angle of attack decreases with rolling, the normal force also decreases. In experiments, however, the static and dynamic cases change in the normal force is found to be different (Levin and Katz 1984). The measurements of Levin and Katz indicate that the time average of the normal force coefficient measured during the roll is smaller than the statically measured values. This difference, for the angles of attack less than 32°, is due to unspoiled vortex symmetry at zero roll angle of for static case, and continuously existing asymmetry for the dynamic case. At higher angles of attack, the vortex bursts occurs earlier for the dynamic case than it happens for the static case which makes the average normal force coefficient for the dynamic case to be 15-20% less than that of the static case.

The measurements made on the delta wing given in Example 3 suggest that during rocking, the oscillatory aerodynamic side force acting on the wing has amplitude of 0.5 and a small phase difference between the roll angle (Levin and Katz 1984). About the rolling characteristics of the delta wing of Example 3, there is detailed information related to experimental results, conditions and comments at 20°, 25°, 30° and 35° angles of attack given in Levin and Katz. On the other hand, the frequency of the normal force coefficient is double the frequency of rolling, and the amplitude changes are in minimum 0.3 and maximum 0.7. In their work, the dynamic values of normal and the sideways forces in terms of static and the roll angle values /max and /av ffi /max/2, as upper and lower limits read,

cndy CNsTC°s/av (8.23a)

CYmaks ^ CNSTsin/maks* (8-23b)

During wing rock the maximum dynamic normal force coefficients, as stated before, can not exceed the values attained in static cases as given by inequality

Eq. 8.23a. On the other hand, according to Eq. 8.23b, the lower limit of the side force acting on the wing is proportional with the static value of the normal force. This means, even for a single degree of freedom problem, there is a minimum sidewise force created. For this reason, in real flight condition wing rock analysis we need to consider the sideways and vertical degrees of freedom in addition to the rolling as shown in Fig. 8.14.

Three degrees of freedom simulations in wind tunnels require building of mobile models which are quite expensive to operate, when possible. This forces us to make measurements in flight conditions and/or to perform detailed numerical simulations to compare the results obtained with single degree of freedom problem (Saad and Liebst 2003). In their work, Saad and Liebst use numerical solution of Euler equations to compute the flight path under the aerodynamic forces computed as three degrees of freedom problem. The kinetic yaw angle can be computed if we take the v as the velocity normal to free stream as follows

Ann = arcsin(v/U). (8.24)

The total yaw angle is determined with addition of angles given by Eqs. 8.22 and 8.24 and. The geometry of the wing consists of a cone with 30° cone angle and a delta planform with 60° sweep. Two different flow case studies were done about this geometry: first study involves only the roll degree of freedom, and the second study was done with three degrees of freedom. The following was observed:

1. The angle at which rocking starts is 5° higher for three degrees of freedom.

2. Roll angle amplitude is 50% for the three degrees of freedom.

3. The sideways motion has 90° phase difference with rolling, which contributes to damping.

4. The occurrence of vortex burst causes the amplitude for three degrees of freedom to be less.

5. The vortex dynamics suggests that the pitching degree of freedom should be included in wing rock analysis.

The comments made above are only based on the numerical solution of Saad and Liebst, and they are not validated by flight measurements.

On the other hand, based on the Navier-Stokes solution, one degree of freedom problem was studied in detail with comparing experimental data given for rolling by Chaderjian and Schiff. Their study is made for the 65° swept delta wing mounted on 8% thick cone-cylinder body. In their study, they consider 15° angle of attack, Reynolds number of 3.67 x 106, Mach number of 0.27, and maximum roll angle of / = 40°. The dynamic roll motion occurs at a frequency of 7 Hz, and the normal force coefficient shows similar behavior to that of experiments while it is predicted little less than the static force coefficient. Another important con­clusion made in their work is the moving of center of pressure towards trailing edge for the dynamic case as compared to the static case. Here, low values of the normal force coefficient could be the reason for the center of pressure to move towards the trailing edge. In addition, the roll moment versus roll angle hysteresis curve is in counter clockwise direction, similar to the case of experimental mea­sures, which indicates that the motion has a damping character. The delta wing left free to roll from the maximum roll angle shows a damping motion experimentally, whereas Navier-Stoke solutions predicts over damping. These computations and experiments are performed at 30° angle of attack, and they converge not to zero roll angles but to half of the maximum roll angle! This shows that if the roll motion is not forced then the delta wing can undergo unsymmetrical damping motion.

The wing rock motion or rolling studied so far is for the slender delta wing whose vortex dynamics is well understood. The non slender wings with round leading edges having about 45° sweep angle at high angles of attack may undergo wing rock for different aerodynamic reasons (Ericsson 2001). The effective angle of attack, as shown in Fig. 8.19b, because of effective yaw angle reads as

Aeff = Л ± arctan(tanrsin/). (8.25)

Here, r is the angle between the roll axis and the free stream direction. The effective sweep angle increases on one side of the wing while it decreases on the other side to spoil the symmetry of the vortices. This causes a net roll moment also on the non slender wings, which may lead to the wing rock. Once the wing rock started, one side of the wing goes up and the other side goes down relative to the root chord. The flow separation becomes possible when the effective angle of attack, Eq. 8.21, is near stall angle. For rolling motion to be continuous and periodic wing rock the negative dissipation is necessary. The necessary negative damping is provided by the ‘moving wall effect’ acting on the boundary layer near the stagnation region. At high angle of attack near the flow separation, the moving wall effect makes the flow to reattach at the uplifting side of the wing and increases the lift at that side while the wing rocks. This way the force increases with the increasing direction of motion to create negative damping. On the other side of the wing which is moving downwards, the moving wall effect increases the separation, which in turn decreases the lift on that side, and naturally reduction in the force and the motion in the same direction create reduction in the damping. Thus, occurrence of negative damping on both sides creates enough energy for wing to rock. This is how the flow induces rocking motion on the non slender wing with round leading edge (Ericsson 2001).

For two different non slender wings both with 45° sweep angle as shown in Fig. 8.21a, b, one with the round the other with the sharp leading edges and lower thickness, we observe a completely different rolling behavior at high angles of attack (Ericsson 2003). The planform given in Fig. 8.21a is a 9% thick delta wing with round leading edge and the other planform is 6% thick with sharp leading edge.

For both types of wings given in Fig. 8.21, the experiments performed at high angles of attack give 50% less rolling moments compared to slender delta wings. The sharp leading edged wing at 20°-25° angles of attack is damped at roll angles of 42° and 0°, respectively when it was left free at 28° roll angle. On the other hand, for the round leading edged wing at 30°-35° angle of attack range, we observe damped motion at zero roll angle which is left to roll at angles /

(b)

= 10° and 30°, respectively. At 25° and 30° angle of attack, the round leading edged wing when left free to roll from 30° roll angle, rocks with 20° roll amplitude at about / = 50°. This means that the undamped rolling motion can be observed experimentally only for the non slender wing with the round leading edge starting from certain roll angles. The observed rocking motion is quasi periodic with an approximate period of 1.5 s. This shows that non slender delta wings with mod­erate sweep have rocking frequency of one order of magnitude less than that of the slender wings with high sweep. This finally proves that the aerodynamic effects causing the rocking of non slender wings occur slower than that of slender wings.

There is a third kind of wing rock occurring at high angles of attack caused by the periodic shedding of the vortices around the left and right side of a fuselage (Ericsson et al. 1996). For an aircraft having slender fuselage with moderately swept wings at high angles of attack, i. e. a > 30° which exceeds static stall angle, we observe this type of wing rock induced by the shedding of vortices from the part of the fuselage which is ahead of the wing. The occurrence of this kind of rocking motion is caused by the vortex shedding from the separated cross flow about the frontal portion of the fuselage. A cylindrically shaped front body rolls about its axis with an angular velocity while it rocks. During this rolling, there is also a vertical flow because of high angle of attack flow separation. Depending on the value of the Reynolds number based on the cross flow velocity there exists a Magnus force, with known magnitude and direction, acting on the cylinder (Ericsson 1988). The Magnus effect on the cylinder is in the positive direction because of the speed of rotation causing the flow is subcritical and laminar. With the increase in the Reynolds number if the critical flow condition is reached, there emerges a Magnus force which is in opposite direction. In flight conditions the wing rock caused by frontal body is observed experimentally at this critical flow regime. When the Reynolds number based on the free stream speed, body diameter and kinematic viscosity is in the range of 1.0 x x 105, the critical

flow conditions are reached. In Fig. 8.22a, b shown is the negative Magnus effect acting on the rotating cylinder in critical flow conditions. The rotational effect on the cylindrical surface causes early transition at the right side of the cylinder, and at the left side the transition is late. The early transition at the right side of the cylinder and reattachment causes a suction force creating negative Magnus force. Meanwhile, from the right side a counter clockwise rotating vortex is shed to the

(b)

wake. This newly shed strong vortex creates a rolling effect which slows down and stops the clockwise rotation, and causes cylinder to rotate in counter clockwise direction. This time at the left side of the cylinder we observe a suction creating a Magnus force directed towards left. That is how the self induced motion feeds itself in creating sustainable wing rock action. In practice, the wing rock caused by the frontal body is the slowest rocking motion with the period of 3.5 s. Here, the flow separation from the moving body and the vortex shedding play an important role in determining the period of wing rock.

Assuming that an axisymmetric frontal body without a tail wing rocks similar to that shown in Fig. 8.22, we can construct the theoretical hysteresis curve for the roll moment versus roll angle as shown in Fig. 8.23a. The ideal curve given in Fig. 8.23a has the negative damping property for the rolling motion; therefore, the wing rock is self sustainable. The ideal curve indicates that as the body rotates in clockwise direction, the roll angle increases to its maximum value, and when the angular speed is 0, the roll angle reaches its maximum value and changes its direction to counter clockwise rotation. Let us denote the time between two suc­cessive vortex shedding as At. Then the counter clockwise rotating body with the increase of negative roll moment goes back to the zero roll position so that in At time interval it starts from 0 roll angle and goes back to 0 roll angle position. In the next At time duration it completes its roll to the left side. Finally, in 2 At time period it completes one cycle of its motion.

In Fig. 8.23b we observe the real version of the wing rock due to vortex shedding from a frontal portion of a fuselage which rocks in -30° and +30° roll

angles. The clockwise direction of the curve near the zero roll region indicates the negative damping while in extreme angles the counter clockwise direction is indicative of positive damping. The difference between the two supplies the necessary energy for rocking.

Classical two dimensional aerodynamics based on the potential theory states that the leading edge suction cancels the streamwise component of the normal force so that there is no drag force acting on the airfoil. In other words, the leading edge suction obtained by the potential theory can overcome any form of drag except the

viscous drag. This makes us wonder if we can make use of the leading edge suction for creating useful aerodynamic forces under special conditions.

It has been observed that the wings with high leading edge sweeps at high angles of attack generate such a high lift that can not be predicted with potential theory. The reason for that is at high angles of attack the vortex generated at the leading edge due to separation merges with the tip vortex to create a strong extra suction force at the upper surface of the wing. This additional lift is called ‘vortex lift’ and it is predicted with the ‘leading edge suction analogy’ by Polhamus by early 1970s (Polhamus 1971). This theory is very much in agreement with the experiments, and it is also called Polhamus theory after its validity was proven on delta wings having low aspect ratio. Now, we are ready to study the vortex lift generation with the aid of Fig. 8.9. According to the potential theory the sectional lift coefficient of a thin airfoil in terms of density, freestream speed and the circulation was given by Eq. 1.1 as I = pUC. We can resolve the lifting force I into its components in the direction of the chord and direction normal to the chord. The normal force is denoted by N, and the force in the direction of chord is called the suction force S. The suction force S, here, is the result of the low pressure zone on the upper surface of the airfoil caused by the fast moving flow. Accordingly, if the angle of attack of the airfoil is a, then the suction force will reads as S = pUTsin a. For a profile with a sharp leading edge even a small angle of attack will cause the flow to separate from the leading edge (Fig. 8.9d). The vortex generated by this separation will create a leading edge suction force S which will now be normal to the surface as opposed to the leading edge suction force of the attached flow.

This leading edge suction force creates the extra lift for the wing. For suction force S to be sustainable, the leading edge vortex must merge with wing tip vortex in a stable and steady fashion. The leading edge vortex shown in Fig. 8.10 merges with the tip vortex to form a reattachment line on the upper surface of the wing which has a stable circulation providing continuous extra lifting force.

Let us use the effective circulation Г and the effective span h of the wing given by Fig. 8.11. If we consider the thrust force T generated by the leading edge vortex, and the adverse effect of the induced downwash wj we get T = pTh(U sin a — wi). Now let us define a non dimensional coefficient Kp related to the potential flow for a wing whose surface area is A as follows

Fig. 8.10 Leading edge vor­tex sheet and the reattach­ment line on the upper surface of a delta wing

Kp — 2Ch/(AU sina).

As the non dimensional thrust force CT of the thrust force T we have

Kpsin2a.

Using the potential lift coefficient Kp we can write non dimensional lift coef­ficient as CL, p = CN, p cos a = Kp sin a cos2 a. Figure 8.11 gives us the relation between the suction force S in terms of thrust force T as S = T/cos Л, where Л is the sweep angle of the leading edge. The vortex lift coefficient CL, v after the leading edge separation reads as

The addition of potential and vortex lift gives the total lift coefficient CL as

CL — Kpsinacos2a + Kysin2acosa. (8.10)

Here: Ky = (1 – ui^)Kp/cosK.

According to Eq. 8.10, at low angles of attack the potential term, the first term of the right hand side, and at high angles of attack the vortex term which is the second term plays a dominant role in the lift. For the low aspect ratio wings at angles of attack lower than 10° Eq. 1.11 gives a lift proportional with a. Similarly, with assumption of small angle of attack we predict the lift proportional with the angle of attack using Eq. 8.10.

In Fig. 8.11b, shown is the spanwise variation of the upper surface pressure coefficient created by suction force S by the separated flow on the upper surface.

In practice, in order to increase the manipulability at high angles of attack using leading edge extension at the root of a wing with moderate sweep creates a new vortex lift (Polhamus 1984; Hoeijmakers 1996). The extra vortex generated by the leading edge extension is shown in Fig. 8.12 with the associated surface pressure coefficient on the upper surface of the wing. Even at moderate angles of attack, the presence of the extra lift contributing to the total is seen in the pressure spanwise distribution. In addition, we observe from the surface pressure curve that the extra vortex creates a strong suction on the upper surface of the wing.

With the aid of Figs. 8.11 and 8.12, we have studied the effects of vortex formation at high leading edge sweep and the leading edge extension of the wings on the spanwise distribution of surface pressure coefficients. Now, we can see the formation of a strong vortex on the total lift coefficient of the wing. Shown in Fig. 8.13 is the total lift coefficient change of a wing with the angle of attack.

The curve in Fig. 8.13 indicated as the potential theory lift curve is obtained by using the first term of Eq. 8.10. In delta wing, on the other hand, both terms of Eq. 8.10 is used in obtaining the lift coefficient curve, where as the leading edge extension lift curve is adopted from Polhamous and Hoeijmaker. In addition, deviation from the Polhamus theory and the experimentally obtained lift loss is shown in Wentz and Kohlman (1971) and Polhamus (1984). Three dimensional wing theory predicts that the induced drag force is proportional with lifting force. The leading edge vortex increases the lifting force, and induces a drag force proportional with the tangent of the angle of attack (Wentz and Kohlman 1971). Accordingly, the total drag coefficient reads as

Cd = cd0 + Cxtana (8-ll)

where CD0 is the drag at zero angle of attack.

In the Chap. 4, the Jones theory provided the lift generated by the cross flow over thin delta wings in incompressible flow. Now, we are going to study the effect of compressibility both in small and large angles of attack. In subsonic flow, depending on Mach number the Prandtl-Glauert formula provides the aerody­namic coefficients (Eq. 1.15a).

In supersonic flows, however, the leading edge sweep angle determines the formula to be used. If the leading edge of the outside of the Mach cone, i. e. supersonic leading edge, the Ackeret formula is sufficient (Eq. 1.20). If it is in the Mach cone, subsonic leading edge, the interaction between the upper and lower surfaces has to be taken into account (Puckett and Stewart 1947). In their study, the leading edge sweep angle Л and the Mach number M are used defining the parameter m as follows:

(Vm2 — 1^ cotK

and for m < 1

– = 2pcot K/E(m’). da

Here, m’ = V1 — m2 and E(m’) is the elliptic integral of second kind (Korn and Korn 1968). If we write Eq. 8.12 suitable for the supersonic flows using Polhamus theory then the potential lift line slope reads as

Kp = nAR/[2E(m’)].

The vortex lift line slope Kv in terms of potential lift line Kp with the aid of Fig. 8.11 becomes

(8.14)

Kv

then for the supersonic

Kv = p[(16 — (AR * b)2)(AR)2 + 16)]1/2/[16 *(E(m’))2}. (8.15)

Here, b = PM2 — 1.

Since the supersonic flow is three dimensional, the induced drag force coeffi­cient for m < 1 and small angles of attack becomes (Puckett and Stewart 1947)

Cdi = aCl[1 – m’/2E(m’)] (8.16)

Example 1 Euler Equation numerical solver for a delta wing with 75° sweep at M = 1.95 and 10° angle of attack gives the normal force coefficient as CN = 0.295 (Murman and Rizzi 1986). Find the normal force coefficient with:

1. incompressible potential flow

2. supersonic potential flow,

3. Polhamus theory.

Solution:

1. For M = 0: we find CN = p/2ARa cos a = 0.289.

2. For M = 1.95: Kp = 2p cot A/E(m’), m = pM2 — 1 cot A = 0.4485, and sin – (1 – 0.4485 / = 63.35, E(63.35) = 1.18 (CRC 1974) Kp = 1.446. This gives CNp = 0.235.

3. According to the Polhamus theory the extra lift created by the leading edge vortex reads as Kv = p [(16 – 1.1487 x 2.8025) (1.1487 + 16)]^/(16 x 1.182) = 2.087, and CN, p = CL, vcos 10 = 0.061. Total normal force coefficient: CN = CN, p + CN, v = 0.235 + 0.061 = 0.296.

Accordingly, the closest solution to Euler’s result is obtained with the Polhamus theory.

Example 2 Find the induced drag coefficient of the wing given in Example 1. at 10° angle of attack.

Solution: CDi = 0.033 from Eq. 8.16.

The viscous and total drag forces acting on the swept wings are given in much detail in Kuchemann (1978).

The extra lift created by the leading edge vortex of a delta wing with small aspect ratio is not sustainable, as shown in Fig. 8.11, when the angle of attack increases beyond a critical value because of the spoiling of the symmetry of the vortex pair. Once the symmetry of the vortex pair is broken, the amount of suction on the left and the right side of the wing are no longer equal, therefore, there emerges a non zero moment with respect to the axis of the wing. This moment causes wing to start a rolling motion which is referred to as wing rock.

For an airfoil pitching in oscillatory motion about a given angle of attack, there is a phase lag between the variation of angle of attack, and the lift in time causing a hysteresis for the lift versus angle attack curve as shown in Fig. 3.9. This phase lag increases and the hysteresis curve becomes more pronounced by increasing as the frequency of oscillation gets larger. This shows us that the response of the airfoil to

the angle of attack change is delayed more and more with increasing unsteadiness. This means, while pitching, although the angle of attack can exceed the critical static stall angle the flow may still remain attached. In case of separation, depending on the size and the location of the bubble, as the lift curve returns back from its maximum, the curve deviates from its normal hysteresis behavior until the bubble reattaches itself. The studies related to this behavior were first seen in 1950s during the experiments performed on the helicopter blades in forced pitch oscillation in forward flight (Halfman et al. 1951; Rainey 1957). In later years, a detailed study by Litva gives the following detailed information on a special profile: (a) The effect of Mach number on aerodynamic damping, (b) the negative damping in large amplitude vertical oscillations, and (c) The maximum normal force is reported to be significant compared to static case. The wind tunnel used by Litva in his experimental studies had the following operational properties: (a) 0.2­0.6 Mach number range, (b) 2.2-6.6 x 106 Reynolds numbers, (c) pitching reduced frequency range 0.04-0.72, reduced frequency range for heaving-plunging 0.04-0.24, and (d) a = 0°-25° average angle of attack range. For an airfoil which had the 13° angle of attack for static separation and maximum normal force coefficient as CN = 1.3, in dynamic tests, on the other hand, pitching about quarter chord point with k = 0.062, M = 0.4 and a = 14.92° at 17° angle of attack, the maximum normal force coefficient was CN = 1.6 (Litva 1969). In Litva’s work on dynamic cases an interesting observation was made on the moment coefficient change which starts at 12° angle of attack before the start of lift loss! On the other hand, the increasing frequency delays both the lift loss and the sudden drop of the normal force while reduces the hysteresis effect. Increase in the Mach number affects both the static and dynamic surface pressure before and after the lift loss in such a way that of the 10-15% chord the pressure increase is more for the sepa­rated flow case. As a result of this, the flow at high subsonic speeds separates at the leading edge and reattaches afterwards; however, the separation occurring near the trailing edge is sustained. The difference between the leading edge separation and the trailing edge separation are given by Ericsson and Reding. The difference for the dynamic case lift increase is the 50% for the leading edge separation and just 15% for the trailing edge separation. The effect of Mach number on the separation is also observed together with the angle of attack increase (McCroskey 1982). In his work, McCroskey describes the trailing edge separation as ‘light stall’, and the leading edge separation as ‘deep stall’.

According to these definitions, the hysteresis curves for a typical airfoil pitching at small reduce frequencies in a low subsonic free stream are provided in Fig. 8.5. These graphs show the sectional lift CL, moment CM, and the drag coefficients CD changes with respect to angle of attack for one cycle of pitching. (a) At the onset of stall, CL preserves its elliptic shape accept near high angle of attack, CM changes in counterclockwise direction with increase in angle of attack, and CD increases slightly by increasing angle of attack. (b) In light stall, the lift coefficient has lost its elliptic character and as it reaches to its maximum value with a sudden drop it goes down by decreasing angle of attack, the moment coefficient behaves normal with increasing angle of attack, after its maximum with decreasing angle of attack

Static case

Fig. 8.5 Dynamic stall: a on set of stall, b light stall, c deep stall it suddenly drops down, goes down in clockwise and climbs up in counterclock­wise manner. The drag coefficient, on the other hand increases by increasing angle of attack to reach its maximum but with decrease in angle of attack the curve even goes down to very small negative region. (c) In the deep stall case, CL curve climbs up to its maximum value and with a sudden and deep drop goes down to its minimum value in clockwise manner, CM curve, before its maximum, first in clockwise then in counterclockwise manner completes its cycle, finally the drag coefficient CD takes quite a large value at the maximum angle of attack and with a sudden drop it goes back to its minimum value as it completes its cycle.

Let us examine the pitching moment change with angle of attack given by a close curve in one cycle. The value of this area, — j> CMda, gives us the amount of work, in non dimensional fashion, done on the flow by the profile in one cycle. If the value of the area is positive, then the profile does work on the flow, which reduces the energy of the profile. This has the damping effect on the profile motion. If the area under the — CM-a closed curve is negative then the flow
performs work on the profile to increase its energy, and that in turn increases the amplitude of the pitching oscillations. The increase in the amplitude of the oscillation caused by the negative damping creates the ‘flutter’. This type of flutter is called ‘stall flutter’. There is a close relation between the stall and the flutter. However, the main difference between the two is that the stall is a more general phenomenon, whereas the flutter is defined as the amplitude of oscillation increase caused by negative damping. (McCroskey 1982). There are some other conclu­sions that we can draw from Fig. 8.5b, c as follows: (b) in negative damping, the drag has a small propulsive effect because of being negative, and (c) in deep stall, the starting of loss in moment before the loss in lift occurs at the angle of attack at which static separation case loss also occurs.

In case of vertical oscillations, for the angles of attack less than the separation angle, the lift curve preserves its elliptic shape with the vertical coordinate h as indicated by the theory. For the angles of attack larger than the static separation angle, during the downstroke of the profile because of the separation, the lift loss takes its maximum value at the lowest position of the profile, and in upstroke of the airfoil because of reattachment the lift increases until it reaches the highest point (Litva 1969). Here, the area under the close curve of the lift coefficient, ^ CLdh, at a moderate Mach number, M = 0.4 and at a low reduced frequency k = 0.068 yields negative damping in one cycle. In this cycle, the moment coefficient change is about -0.1. The earlier experiments conducted did not report any negative damping (Halfman et al. 1951; Rainey 1957). The reason for this is because Litva has worked at higher Mach numbers and at higher amplitudes. A further increase in the Mach number reduces the negative damping eventually to zero because of creation of local shocks at critical Mach number of the profile.

Although the dynamic stall phenomenon experiments and visualizations are helpful for obtaining useful empirical relations, see Problem 8.9 (Ericsson and Reding 1980), it seems more detailed and robust analyses are necessary for engineering applications (McCroskey 1982). The Computational Fluid Dynamics (CFD) as a tool gives this robust and detailed information with numerical solutions of Navier-Stokes equations. The pioneering work on the dynamic stall study of a pitching oscillation of NACA 0012 airfoil at Reynolds numbers of 5,000 and 10,000 with reduced frequencies of 0.50 and 0.25 was done by Mehta (1977). The agreement with the experimental work of Werle and the work of Mehta was the early indicative, in those years, of the success of CFD as an analysis tool. The aforementioned work required extensive computational time for one cycle of computations, therefore, especially for the flows with high Reynolds number and the turbulent flows necessitated faster and more efficient codes to reduce the computation times to reasonable levels by means of zonal methods described before (Wu et al. 1984). The integro-differential method developed by Wu and Gulgat, is implemented by Tuncer et al. (1990) for the dynamic stall analysis of NACA 0012 airfoil at Reynolds number of 106 at various reduced frequencies to compare with the experimental work (McCroskey 1981). As a turbulence model Baldwin-Lomax model which is applicable for separated flows also, is implemented. The numerical solution of the Navier-Stokes equations, Eq. 8.2, together with Eq. 8.9 was performed in moving coordinates using the effective viscosity. The angle of attack changed as a(t) = amin + (amax — amin)(1 – cos xt)/2. Shown in Fig. 8.6a is the instantaneous streamlines for k = 0.15, and amin = 5°, amax = 25°, for pitch oscillations of the airfoil. Before the oscillation is started at t = 0, the steady state solution is obtained for a 5° angle of attack. During upstroke, although the static stall angle is exceeded up to angle of attack being 20° flow does not separate. After 20°, however, flow separation starts from the trailing

edge and moves toward the leading edge and it reaches to leading edge at 23°. At about 23.9°, at quarter chord a leading edge vortex is formed and it covers the entire upper surface at 24.9°. Going back from the maximum angle of attack of 25°, the vortex leaves the airfoil surface, and moves downstream with a speed of

0. 3 U?, whereas the experimental value of this speed ranges in. 0.35-0.40 U?. During downstroke, the vortex starts to separate from the leading edge at 22.8° angle of attack, and with decreasing angle of attack the flow starts to reattach to upper surface similar to that of potential flow until the minimum angle of attack is reached.

Shown in Fig. 8.7a, b are the numerical and experimental values of upper surface pressure coefficient variations during the upstroke and the downstroke motion of the airfoil. As the angle of attack increases, the suction effect of the leading edge vortex and the pressure increase after the separation of vortex from the surface are easily seen, and during the downstroke the flow reattaches, but because of thick boundary layer formation only at the minimum angle of attack the potential surface pressure distribution can be reached, Fig. 8.7.

In Fig. 8.8a shown are the numerical and experimental (a) lift, (b) drag, and (c) moment coefficient plots for the same airfoil at k = 0.15 for one cycle of motion at which both experimental and numerical results show the same trend. (a) The lift increases with the increase of angle of attack until reaching maximum, and as the angle of attack becomes smaller the vortex separating from the surface causes lift to drop suddenly. At 22° angle of attack, the new surface vortex forms to increase the lift, however, because of the thick boundary layer formation the lift still drops down until 9° angle of attack and it takes its potential value when the angle of attack becomes minimum. (b) The drag coefficient, on the other hand, increases

with increasing angle of attack, after the static separation angle it gradually increases until 23° at which there is a sharp increase to CD = 1.0′ at a = 25°. During downstroke, the drag drops down to values even lower than the values attained at angles of attack equal to that of at upstroke angles, and finally it reaches 0.0 at minimum angle of attack. It is interesting to note that the drag becomes slightly negative when a 10° during the downstroke, which indicates that there is a slight propulsive force. (b) The moment coefficient stays at its zero value as the angle of attack increases until the leading edge suction occurs. Afterwards, the moment becomes negative because of growth of the vortex and its streamwise movement which makes CM = -0.6. Returning from maximum angle of attack, moment increases gradually as the angle is in 23°-19° range it decreases again but then it starts to increase to its maximum value of 0.1, and decreases to 0. The moment coefficient is in agreement with the experimental results. However, the area under the — CM-a curve gives a negative value, larger in magnitude than the value obtained numerically, for the experimental measurements, meaning that the flutter is reached before that predicted numerically.

In Fig. 8.8b shown are the lift and moment coefficient variations with respect to angle of attack in pitching with k = 0.10 and k = 0.25. For all three reduced frequencies the lift coefficient curves show similar behavior; however the moment coefficients have a tendency to give negative damping with increasing reduced frequencies.

The detailed CFD analysis of the dynamic stall phenomenon has been given here. In Leishman, however, extensive summary of empirical models introduced earlier are given. In addition, the effect of sweep on dynamic stall of a wing is studied with 30° angle of sweep at M = 0.4 free stream Mach number (Leishman). In his study Leishman observed that the lift curve slope is not affected with sweep; however, the separation angle increases about 4°, and during the downstroke the hysteresis curve gets narrower. In the moment diagram, the moment becomes more positive with sweep and the area under the hysteresis curve tends to give more negative damping.

In small sweeps for a finite wing the measurements made at k = 0.1 and M = 0.2 as the sweep increases:

1. lift curve slope decreases,

2. static separation angle of attack increases,

3. the lift curve hysteresis gets narrower,

4. the angle of attack at which the moment loss occurs is getting bigger.

A considerably more simple way of studying unsteady airloads at high angles of attack is possible via state-space representation of aerodynamic characteristics based on an input state variable (Goman and Khrabrov 1994). The sectional lift and moment coefficients of an airfoil undergoing an arbitrary unsteady motion can accurately be determined using the static tests separation point movement pre­scribed as an input state variable (see Problem 8.13).

At large sweep angles the separation phenomenon has a different character than the things happening at small sweeps. Let us next study what happens at high sweeps as the flow separates from the leading edges which may be sharp or round.

The books on classical aerodynamics depict the picture of static stall as the sudden lift lost after a critical angle of attack. The aerodynamic aspects after the static stall are not usually emphasized (Abbott and Von Doenhoff 1959). However, even in early 1930s there were some experimental studies performed on different profiles to predict their lifting characteristics beyond stall (Eastman, Anderson). On the other hand, in later years both improvement in visualization and measurement techniques, and numerical solutions of Navier-Stokes equations in advanced computational means and tools enable researchers to study flow separation and corresponding lift lost at least with laminar flow studies (Mehta 1972). The whole flow field was solved by finite difference in Mehtas’ study which required very large computer memory and time in those days. In addition, solving the attached and separated flow regions with full Navier-Stokes solver caused extra numerical errors. This type of errors and computational time were reduced by means of an integro-differential method (Gulcat 1981, 2009a, b Wu and Gulcat 1981). The integro-differential method reduces the computational time with increasing Rey­nolds number (Wu et al. 1984). Now, we can step by step show the solution domain obtained by integro-differential method for one cycle of flow features by means of instantaneous streamlines of the separated flow past 9% thick Joukowsky airfoil at 15° angle of attack in Fig. 8.2a-j. The laminar flow is studied at Reynolds number of 1,000 with the initial conditions given at t = 0 as the non circulatory potential flow solution whose streamlines are shown in Fig. 8.2a. Here the non dimensional time is given by the free stream speed and the chord length of airfoil. After the impulsive start, for a short time the flow continues without separation as shown in (b), and as seen in (c) near the leading edge a separation bubble appears. This bubble grows larger to become the main bubble as shown in (d), and forms a large clockwise vortex covering almost the entire upper surface. The main bubble afterwards bursts and separates from the upper surface as shown in (f). The sep­aration of the main vortex from the surface and its movement towards the wake with the main stream generates a counterclockwise rotating vortex at the trailing edge and a secondary weak bubble at the upper surface (f, g). While the secondary surface bubble grows as shown in (g), the trailing edge vortex detaches from the trailing edge and gets carried into the wake (g, h). Meanwhile, recently generated leading edge separation bubble spreads over the entire upper surface in place of the weakening and bursting secondary bubble (h). Thus, one cycle of events becomes complete, starting at t = 1.89, and ending at time t = 7.41 to have the non dimensional period of T = 5.52. The Strouhal number for this flow hence becomes 0.18. The flow separation and one period of vortex formation appear to be unsteady although the boundary conditions of the flow remain the same. Now, we can observe the time variation of the lift and the drag coefficient by examining Fig. 8.3. Impulsively started airfoil at t = 0+ has no circulation and therefore, it has zero lift but very large drag. For this reason, Fig. 8.3 shows the lift and the drag curves together which are started for t > 0 just before the flow separation.

The lift coefficient CL reaches its maximum value when the separation bubble covers the entire upper surface at t = 1.89. This means the suction created by the separation bubble generates additional lift. However, after the bursting of the main separation bubble, the lift coefficient drops down from 1.3 to its minimum value

0.2 in the time interval t = 2.93 to t = 5.69. After reaching its minimum value, the lift coefficient increases slightly as the new separation bubble grows and covers the upper surface eventually.

The drag coefficient, given in Fig. 8.3, changes by time similar to the lift coefficient with a phase difference. Since the Reynolds number is low, the drag coefficient values are higher than usual. The drag coefficient takes its maximum value 0.35 when the main separation bubble covers the upper surface causing the largest suction force normal to surface whose streamwise component is quite high. After the bursting of the main bubble, the drag value drops down to its minimum value of 0.15. The growth of new leading edge bubble causes suction to increase, and this in turn makes drag grow to 0.2. As shown in Fig. 8.3, the results of the zonal method is in agreement with the results of the full Navier-Stokes solver given as reference (El-Refaee 1981).

The flow separation causing static stall is a strong separation from the upper surface under constant high angle of attack. The analysis of separated flow regions is possible with numerical solution of Navier-Stokes equations. At the lower surface of the airfoil boundary layer is formed because of favorable pressure gradient. This enables us to divide the entire flow region into two different regions with different flow features, that is attached and detached flow regions which are connected to each other. In the attached flow the boundary layer equation is solved, and Navier-Stokes equations are employed in detached region to give fast and accurate results. These two different regions are interlaced with an integral approach, and the conditions at infinity are satisfied while computing the vortex sheet strength on the surface of airfoil. For this purpose let us express the gov­erning equations in velocity-vorticity formulation in two dimensions. The defi­nition of vorticity: x = V x V from the velocity field V gives us the continuity and the momentum equations as follows:

VxV = 0

and,

0Ю ~

= — (V xV)x + vV2x.

The boundary layer approximation gives the relation between the vorticity and the velocity component vs parallel to surface as follows,

Here, n is the normal direction to surface, and

(8.5)

At a station along the boundary layer, if we know the vorticity value, then integration of Eq. 8.5 in the normal direction gives us the velocity component parallel to surface as follows

n

vs(s, n) = -^ x(s, n)dn. (8.6)

0

Knowing vs component in two consecutive stations let us utilize the integral of continuity equation in normal direction to obtain

Equation 8.5 is solved to obtain the vorticity values at the new time level t by forward differencing in time, forward differencing in s and central differencing in y direction. This gives us a tri-diagonal system of equations for new time level vorticity values to be found at a given station (Wu and Gulcat 1981). As the boundary conditions of Eq. 8.5, the vorticity at the edge of the boundary layer is taken as zero, and the surface vortex sheet strength computed by integral approach is utilized as the surface boundary condition. We can make use of the continuity of vorticity to obtain an expression for the induced surface velocity by the velocity field at infinity and the vorticity field excluding the surface of the airfoil. This gives us the following integral relation

In Eq. 8.8, S+ is the neighborhood of the profile surface S, R is the vorticity field, rs is the point on the surface, ms is the vorticity value at the surface, and V(rs, t) is the time dependent surface velocity vector. Once the surface velocity and the free stream velocity are described, together with the known vorticity field from Eqs. 8.3 and 8.5 we can obtain the surface vortex sheet strength from Eq. 8.8. The kinematics of the separated flow region can be formulated in terms of the stream function W and the vorticity ю as follows

V2W = – ю. (8.9)

The kinetics of the separated flow on the other hand is given by Eq. 8.3. The simultaneous solution of Eqs. 8.3 and 8.9 with finite differencing gives vorticity and the velocity fields. The Integro-differential method applied to a flow past an airfoil at high angle of attack can handle the three different flow regions
simultaneously as shown in Fig. 8.4. These regions are (a) ideal flow region, (b) boundary layer region, and (c) the separated flow region.

The ideal flow region has zero vorticity; therefore, we only need to have the farfield boundary condition effective on the body as the contribution to the flow field from this region. The viscous region, on the other hand, induces velocity according to the Biot-Savart law. The induced velocity can be expressed as the vortex sheet strength on the airfoil surface with the aid of Eq. 8.8. After the impulsive start, the diffused vorticity covers a small flow region around the airfoil surface which is the agent that creates the vorticity. By time this vorticity is convected in the flow direction to increase the vorticity field. We initially take the computational domain small and enlarge it in parallel with the size of the vortex region. This enables us to keep the computational work minimum, and saves us from having spurious errors because of unnecessarily performed computations.

Studying the leading edge separation from certain profiles in two dimensions helps us to understand the nature of the suction generated by the vortex before it bursts. In three dimensional flows because of high sweep we can have the vortex to roll rather than to burst in sustaining the suction force to provide the extra lift at high angles of attack which may cause the 2-D profile to stall.

Now, we can study another stall phenomenon which is dynamic stall of pitching or plunging airfoils in periodic motion wherein the onset of leading edge flow separation is delayed in terms of angles of attack higher than the occurrence of static stall for the same airfoil.

Most of the material we have studied so far in general are the topics belonging to classical aerodynamics related to flows past thin or slender objects in small angles of attack for the purpose of generating lift. After the 1970s we see that the boundaries of classical aerodynamics are crossed because of advances made in computational as well as experimental techniques. The flow field analysis of low aspect ratio wings with high swept leading edges at high angles of attack enabled researchers to predict the extra lift generated because of leading edge separation which is exactly the case for some of the biological flows in nature. Utilization of leading edge separation helped aerodynamicists to design highly maneuverable military aircrafts to be used for military purposes. As is known from the classical aerodynamics, the leading edge separation from the wings with a little sweep or no-sweep, on the other hand, causes lift loss. This type of wing must have high lift while cruising at a constant speed and during landing with low speeds must have even higher lift without stalling. Other­wise, unsymmetrical lift loss, either from the left or right wing creates a rolling moment about the axis of the plane, and this causes it to rock. The larger roll moments about the axis of the plane cause spin (Katz and Plotkin 1991).

Since it is not possible to solve the separated flow fields with analytical methods, we have to resort to experimental measurements and visualization techniques or to numerical methods. The numerical and the experimental methods are used together in complementary fashion for the analysis of separated flows to predict the aerodynamic characteristics of relevant configurations. In this chapter, first we are going to study in a detail the flow separation around an airfoil at a constant angle of attack. The sudden lift loss at a constant high angle of attack because of leading edge separation is called static stall. The numerical as well as experimental studies about static stall will be given. Afterwards, the flow separation and the related lift loss at variable angle of attack which is called ‘dynamic stall’ are going to be analyzed in detail. Finally, three dimensional analyses of swept wings with extra lift created by leading edge sepa­ration will be studied. The summary of leading edge flow separation and the vortex sheet formations are shown in Fig. 8.1 for both swept and unswept wings. According to Fig. 8.1a (adapted from Katz and Plotkin 1991), for an unswept wing with high

U. Gulfat, Fundamentals of Modern Unsteady Aerodynamics, 245

DOI: 10.1007/978-3-642-14761-6_8, © Springer-Verlag Berlin Heidelberg 2010

aspect ratio there is a two dimensional separation as indicated with A-A cross section. In this two dimensional separation, the clockwise vortices leaving the leading edge and the counter clockwise vortices leaving the bottom surface at the trailing edge form a periodically formed vortex street at the wake of the wing. If h is the vertical distance between the centers of clockwise and counter clockwise vortices, f is the frequency of vortex generation, and U is the free stream speed then the Strouhal number for average Reynolds numbers reads as (Katz and Plotkin 1991)

fh

St = U ^ 0.1 – 0.2. (8.1)

The wings with moderate sweep at their leading edges have the flow separation with more than one vortex at each side of the wing as shown in Fig. 8.1b. Shown in Fig. 8.1c is the wing with the high sweep, K > 70°, which generates a pair of very strong vortices to roll up immediately after separating from the sharp leading edge, section BB. These high strength vortices generate suction at the upper surface which in turn creates additional lift. The pair of counter rotating vortices generated by the leading edge separation at higher angles of attack of the highly swept wings tends to have their symmetric strength uneven. This causes a rolling moment about the axis of the wing. Initially the rolling moment is small and periodic in nature; therefore, it causes the wing to rock. Further increase in angle of attack causes sudden burst of one of the vortices. This puts the wing in spin. Both wing rock and spin are the unsteady motions induced by the flow.

The periodic heaving and/or pitching motion of an airfoil, as a forced oscilla­tion, is for long known to be the major source of thrust generation for flapping wings. During this type of oscillations the flow separation takes place at a larger angle of attack than the angle at which the static stall occurs. The value of reduced frequency of oscillation plays an important role in determining the lower limit of the angle of attack at which the dynamic stall takes place. Now, starting from static stall let us see the unsteady aerodynamic aspect of the phenomena which can be included in modern subjects.

Hitherto, we have seen the various properties of hypersonic flow past different type of body shapes. Now, we can apply this information to analyze the hypersonic aerodynamics of an aerospace plane. The concept of sustainable hypersonic flight of a vehicle having most of the features described in previous sections and riding on the pressure created by the shock surface under its body formally was intro­duced in late 1950s (Nonweiler 1959). In his work, Nonweiler proposes inverted W and V shaped cross sections for delta type wings with surface streamlines and lower surface shocks as shown in Fig. 7.25. The weak shocks appearing at the lower surface create sufficient lower surface pressure so that the lift generated is adequate for a sustainable flight.

This type of aerospace planes which makes use of the lower surface pressure is also called ‘waverider’. According to Nonweiler’s analysis an aerospace plane which was named ‘karet’ then, has stagnation temperature of 1,150°C, at the mid chord region the temperature drops down to 500°C at 80 km altitude with free stream speed of 6.5 km s-1.

Fig. 7.25 Inverted, a W and b V shaped delta wings and shocks and surface streamlines

The waverider concept was quite popular until 1970s, and lost its popularity for some time because of its low lift to drag ratio, L/D. Thanks to numerical opti­mization techniques, towards the end of 1980s, the waverider gained popularity again (Bowcutt et al. 1987; Anderson et al. 1991). In waverider studies, the viscous effects were also considered for the evaluation of optimum (L/D)max for various free stream Mach numbers, at different altitudes and even for planets whose atmospheric properties are known. The previously known (L/D)max barriers were broken with numerical optimization techniques. The free stream Mach number figures given by Kuchemann for 5 M 10, can be improved approximately 1.5. As shown in Fig. 7.26, for a wide range of free stream Mach numbers (2 M 25), Kuchemann’s previously given (L/D)max = 4 (M + 3)/M barrier curve has been changed to (L/D)max = 6 (M + 2)/M because of optimization. Bowcutt et al. even exceed the second curve shown in Fig. 7.26 for M = 20. However, for M = 25 the optimum L/D value is below the given curve. The reason for this behavior is attributed to the Reynolds number effect at high altitudes. The Reynolds number for both free stream Mach numbers is laminar; therefore, for M = 20 viscous drag becomes low to give higher (L/D)max. This gives deviations from the averaged behavior for predicting (L/D)max values.

In order to find the optimum shape of a waverider at a given free stream and altitude, first an axially symmetric flow with a weak attached shock of a cone whose angle is the Mach cone angle of the given free stream Mach number is considered (Example 7.3). The leading edges of the waverider are placed inside the conical shock as shown in Fig. 7.27. The upper surface of the waverider is configured as a cylindrical shell surface parallel to free stream. The bottom sur­face, on the other hand, consists of a base curve tangent to the Mach cone surface, and it joins to the leading edge with a bell shaped curve as shown in Fig. 7.27.

After determining the lower surface pressure created by the shock, and the corresponding lifting pressure, we can calculate the total lift by integrating the lifting pressure over the surface. The total drag then is computed by the wave drag induced by the thickness of the wing and the viscous effects. The surface shape

Fig. 7.27 Waverider geome­try placed in shock cone

conical shock surface

which makes the total drag minimum gives us the maximum (L/D) ratio. Calcu­lating viscous drag with reference temperature method, because of its simplicity, is preferred over the boundary layer integral methods (Anderson et al. 1991). The reference temperature method can give the Reynolds number in transition to turbulence based on the free stream Mach number together with the viscous drag in turbulent flow (Anderson et al. 1991).

The waverider shapes based on the ideal flow analysis as given above were also analyzed by solving the Euler Equations (Jones and Dougherty 1992). In that study, the surface pressure distribution is found to be in good agreement with Bowcutt solution and the experimental results for free stream Mach numbers of 4 and 6. In order to find the matching results, a special grid generation based on the adaptive meshing around the sharp leading edges was used.

The optimum shape and the flow field around a waverider was also studied together with Navier-Stokes and Euler solvers results compared (Takashima and Lewis 1994). In their study, Takashima and Lewis found agreement with the surface pressure solution obtained with Navier-Stokes and Euler solvers at free stream Mach number of 6, Fig. 7.28. Here, in that study, the difference between

Fig. 7.28 Upper and lower p / p pressure of a waverider at

2

1.5

y/b

the lower and upper surface pressures gives us the lifting pressure quite similar to that of ideal solution except around the leading edges. While obtaining the viscous solution the flow field is discretized with a fine grid near the surface where the first point above the surface has its y+=0.1 and the sharp leading edge is a little bit smoothed to have a stable solution. The rounding of a leading edge has an effect on the (L/D)max value. It has been observed that if rounding is made with leading edge radius ratio to length of the waverider less than 0.1% then L/D remains constant, if it is larger than 1% than L/D value decreases (Lewis and McRonald 1992).

In their study, Lewis and McRonald extend their work to the aerodynamic analyses of such waveriders which exceed Mach numbers of 50 while passing through the atmosphere of other planets to make use of a gravity assists in their sustainable hypersonic flight.

Approximately 100 km above sea level in the Knudsen number range of 0.05 < Kn < 1 the continuum hypothesis is no longer valid. Therefore, the optimum waverider analysis is made with Monte Carlo method which gives very low L/D ratios (Rault 1994). In his study, based on the Monte Carlo Method, Raul find L/D = 0.197 at 105 km altitude, free stream Mach number of M = 25, and Knudsen number of Kn = 0.05.

The experimental and theoretical work performed so far indicates that towards the end of the first of quarter of twenty-first century, the nations or union of nations with advanced technology will probably start the manned or unmanned sustainable hypersonic flight with prototype waveriders.