Acc. Sc. Torino - Atti Sc. Fis. 136 (2002), 49-58, 5 ff.

MECCANICA DEI FLUIDI

Aerodynamic Constraints for Vortex Trapping Airfoils

Nota di BERNARDO GALLETTI, ANGELO IOLLO e del Socio corrispondente LUCA ZANNETTI presentata nell’adunanza del 12 Giugno 2002

Riassunto. Si mostra come `epossibile ricavare la geometria del profilo capace di catturare un vortice che realizza una data distribuzione di velocit`a, essendo tale distribuzione di velocit`aassegnata in funzione dell’ascissa curvilinea fissata sul contorno del profilo. Vengono altres´ı espressi i vincoli che detta distribuzione deve soddisfare affinch´eil prob- lema risulti ben posto.

Abstract. The inverse problem for a vortex capturing airfoil is dealth with. A design velocity distribution is given as a function of the curvilinear abscissa of the airfoil contour. The velocity distribution constraints that lead to a well posed problem are found.

Aerodynamic bodies have certain architectures and shapes that are the re- sult of the progress that has been made in understanding the flow physics. For example, the discovery of the mechanism of lift has led to sophisticated wing designs of minimum induced drag. By the end of the sixties, improve- ments in the understanding of compressible flows led to the design of shockless airfoils at supecritcal Mach numbers [1]. Today, the advances in aircraft per- formance concern the scrutiny of transition and turbulence phenomena, but the mechanism that will grant a break-through in delaying transition or re- ducing turbulent drag will probably be active rather than passive. By active mechanism we mean an artificial modification of the flow which isinducedby the actual flow status, therefore based on sensors and actuators which are able of an automatic reaction. In this context we have studied, using the inverse boundary value problem within potential flow theory (see for example [2]), the design of new airfoil shapes that might become of some interest thanks to active control. An airfoil is considered with a cavity of such a shapethataregionof vortical flow is trapped inside the cavity. Ideally, it represents a virtual flap that provides high lift with no mechanical part being involved. In particular, there are equilibrium configurations of a point vortex that satisfy the Kutta condition at the trailing edge both for a flat plate, see[3], and for the Joukowski airfoil [4]. Alternately, a trapped vortex can be used to let the flow overcome the regions of unfavorable pressure gradient. In [5] it is shown how such airfoils should be designed to allow the development of a cyclic boundary layer inside the cavity. It is also shown that the correct high Reynolds number limit of the flow in the cavity is the Batchelor model, i.e., a region of constant vorticity. These ideas are not so far as they might seem from application.TheEKIP aircraft, see [5], is a bluffbody that hosts a turbofan which allows an attached flow over its upper body surface by a series of vortex trapping cavities. The vortices are stabilized using passive suction. In practice,thecapturedvor- tices result to be highly unstable due to upstream receptivity and secondary separations inside the cavity. We have already dedicated work to the active stabilization of such flows [6] [7]. In this paper attention isconcentratedona related problem, namely, the solution of an inverse aerodynamic problem for vortex trapping airfoils. The inverse problem in aerodynamics has its roots in the worksofMangler [8] and Lighthill [9] who solved the problem of determining anairfoil,given the velocity distribution over its surface. Lighthill discovered that the velocity distribution had to satisfy certain integral constraints inordertoobtainclosed airfoils and to verify the given velocity at infinity. Such conditions go under the name of solvability conditions. The inverse problem formulation for a vortex trapping airfoil involves additional constraints onthedesiredvelocity distribution. Cavities that allow the formation of a stable vortical region are in particular sought. To this end, the vortical region inside the cavity is modeled by a single point vortex. Such a model is approximated as it concentrates all the vorticity of the cavity in one point. Experimental results aswellastheoretical results [5] show that the vorticity is mostly constant in the cavity except in the boundary layer, according to the Batchelor model. Nevertheless, as we show in the following sections, the point vortex model allows the expression, in closed form, of the additional constraints on the prescribed velocity distribution. Thus we obtain an airfoil which generates the given velocity on itsboundaryand sustains the presence of a vortex in stable equilibrium. We heavily rely on the techniques presented in [2] and [10]. The well known inverse boundary value problem for a single isolated airfoil, is briefly presented in the next paragraph to make the ideas and the terminology clear. Then we give the main result of this paper, namely the constraints that have to be satisfied by the prescribed velocity on the airfoil to obtain avortexcapturing airfoil. Additional details are found in [11]. Simple airfoil Let us represent the airfoil on the complex plane z = x + iy (fig. 1(a)). The airfoil contour Lz is closed. The trailing edge T is sharp. It is located on the axis origin and the value of its external angle is given as επ,withε ∈ [1, 2] with ε "=1.Theairfoilchordformsanunknownangleα with the x-axis, while the asymptotic flow velocity v∞ has the same direction of the positive x-axis. Acurvilinearabscissas runs clockwise along Lz,withtheorigins =0located y v (z)

Gz τ vp>0 v b Lz s* s=L A v p<0 πε v α T x 0 s* L s s=0

−v (a) (b)

Figure 1: (a) z-plane. (b) Design velocity distribution (dotted line for cusped trailing edge).

η ψ v (ζ) (w) Lζ c<0 Gζ Gw Γ γ* (s=L) T T A β 1 ξ T ϕ0 ϕ1 ϕ β (s=0) i β u0 e A λ vc>0

(a) (b)

Figure 2: on the trailing edge T .ThecontourlengthL is given. The unit vector tangent to the contour is denoted byτ ˆ. The inverse problem consists in finding the airfoil shape for agivenfree stream flow velocity v∞ and a given velocity distribution vp = vp(s)onthe airfoil contour Lz (fig. 1(b)). The solution is obtained by finding the conformal mapping z = z(ζ)thatmapstheunitcircleoftheζ-plane (fig. 2(a)) onto the airfoil of z-plane once the complex potential pertinent to the given velocity distribution has been determined under the constraints limζ→∞ z(ζ)=∞ and z(1) = 0. Let w = w(z)=ϕ(x, y)+iψ(x, y)bethecomplexpotential.Sinceψ =cost on an impermeable solid wall, we assume ψ =0ontheairfoilboundaryLz. Moreover, we are free to set ϕ(s∗)=0,withs∗ denoting the curvilinear abscissa of the stagnation point (fig. 1(a)), so that

s ϕ(s)= v(s)ds. (1) !s∗

By setting ϕ1 = ϕ(L), ϕ0 = ϕ(0) it follows that the airfoil circulation is Γ=ϕ1 − ϕ0,withΓ> 0whenclockwise. According to the invariance of complex potential for conformal mapping w(ζ)=ϕ + iψ = w(z(ζ)). The expression of w(ζ)isknown:

" " " u eiβ " Γ w(ζ)=u e−iβ ζ + 0 − ln ζ + c (2) 0 ζ 2πi 0 " −iβ where u0e is the complex velocity dw/dζ at infinity and c0 is a real constant. Let ζ = r exp(iγ), therefore " ∂ϕ Γ vc(γ)= = −2 u sin(γ − β) − (3) # ∂γ $ 0 2π " r=1 is the velocity distribution on the unit circle contour relevant to the given velocity distribution vp(s)ontheairfoilcontour. ∗ The values of u0 and β are determined by enforcing the conditions: ϕ(γ )= ∗ ∗ ϕ(s )=0,ϕ(0) = ϕ(L)=ϕ1, vc(γ )=0andvc(0) = vc(2π)=0,thatis: " " Γ 2 u cos(γ∗ − β) − γ∗ + c =0 0 2π 0 2 u0 cos β + c0 = ϕ1 Γ −2 u sin(γ∗ − β) − =0 0 2π Γ 2 u sin β − =0 (4) 0 2π ∗ which are solved with respect to u0, β, c0 and the stagnation angle γ . The function s = s(γ)relatingtheunitcircleangleγ to the airfoil arc length s is implicitly expressed by the equation ϕ(s)=ϕ(γ)anditsderivative % % % s (γ)canbedeductedbyvp(s)=ϕ (s), vc(γ)=ϕ (γ)andeq.(3),thatis " Γ " γ γ % −2 u0 sin(γ − β) − −4 u0 sin cos − β % ϕ (γ) 2π 2 # 2 $ s (γ)= % = = (5) ϕ (s) vp[s(γ)] vp[s(γ)] " which is a continuous negative function for 0 <γ<2π and which has a zero of order ε − 1forγ =0andγ =2π. The determination of the logarithm of the mapping derivative

dz dz dz log =log + i arg (6) dζ %dζ % #dζ $ % % % % is obtained according to the classical% solution% of the problem of finding an analytic function whose real part

dz log =log|s%(γ)| . (7) %dζ % % %|ζ|=1 % % is known on the unit circle contour.% % The function log(dz/dζ)hasalogarithmicsingularityinζ =1whichcan be avoided by considering the function

dz 1 ω(ζ)=P (ζ)+iQ(ζ)=log − (ε − 1) log 1 − (8) dζ # ζ $ whose real part is known as well γ P (γ)=ln|s%(γ)|−(ε − 1) log 2sin . (9) & 2 ' According to the Schwarz formula, for the region exterior to the unit circle

1 2π eiτ + ζ ω(ζ)=− P (τ) iτ dτ + iQ(∞)(10) 2π !0 e − ζ with Q(∞)=arg(dz/dζ)∞ = −β.Byintegratingeq.(8)weobtaintheinverse problem solution ζ ε−1 " ω(ζ ) 1 % z(ζ)= e 1 − % dζ . (11) !1 # ζ $ The previously described problem is in general ill-posed forthevelocityat infinity v∞ and the airfoil velocity distribution vp(s)cannotbearbitrarychosen but have to satisfy three constraints, as found by Lighthill [9]. The first constraint expresses the compatibility between vp(s)andv∞,and is found by considering that the mapping derivative at infinity has to satisfy (dz/dζ)ζ→∞ = u0 exp(−iβ)/v∞,thatis,accordingtoeqs.(8),(10)

1 2π u P (τ)dτ =log 0 . (12) 2π !0 v∞ Other two constraints are expressed by the airfoil closure conditions. For the airfoil to be closed, the integral (dz/dζ)dζ taken along the unit circle contour must be zero. Since dz/dζ is regular( in the domain exterior to the unit circle, its residual at infinity is null. According to eqs. (8),(10) this condition yields

2π 2π P (τ)cosτdτ = π(ε − 1), P (τ)sinτdτ =0. (13) !0 !0 In the following section it will be shown that a quasi-solution can be obtained using a variational approach, where by quasi-solution we intended the solution for a modified velocity distribution that satisfies the constraints and is the closest to the original one according to a given criterion. Vortex Trapping Airfoil Examples of airfoils that are capable of trapping a vortex in stable equilib- rium are described in [5] [7]. As shown in fig. 3(b), these airfoils are character- ized by a sharp edge A in addition to the trailing edge T .Theflowseparates at A and a vortex is trapped in the subsequent cavity. The solution method of the inverse problem is the same as that shown in the previous section. For a given velocity distribution pertinent to a captured vortex (fig. 3(a)), the solution is found by determining the function z = z(ζ) that transforms the unit circle of the ζ-plane (fig. 3(c)) onto the airfoil of the v

vT vA

0 sB sA sD L s

−vA

−vT

(a)

y η K K z πεA 0 (z) (ζ) A D ζ D v v <0 v Gz 0 c<0 >0 p p>0 G vp >0 ζ b L vc B z ζ s=L 0 τ vp<0 A −K (s=L) πε v α T Γ+K T

T x 1

<0 β ξ c

s=0 v (s=0) i β u0 e B λ Lζ vc>0 (b) (c)

Figure 3: (a) Design velocity distribution (dotted lines for cusped edges). (b) z-plane. (c) ζ-plane. z-plane (fig. 3(b)). The meaning of the notations and symbols that have been used are the same as in the previous section and can be deductedbyfig.3 The complex potential w(ζ)=w(z(ζ)) is:

iβ −iβ " u0e K ζ − ζ0 Γ+K w(ζ)=u0e ζ + − ln − ln ζ + c (14) ζ 2πi ζ − ζ0 2πi " ∗ where ζ0 =1/ζ ,Γisthetotalcirculation,K,ζ0 are the circulation and the location of the trapped vortex, respectively, and c is a complex constant. The complex velocity is therefore

iβ dw −iβ u0e K ζ0 − ζ0 Γ+K = u0e − − − (15) dζ ζ2 2πi (ζ − ζ )(ζ − ζ ) 2πiζ " 0 0 The flow past the circle has four stagnation points, two are located on the edge images A, T ,theothertwoaredenotedbyB and D,asshowninfig.3.The s value Γis provided by Γ= ϕ(L) − ϕ(0), with ϕ(s)= sA vp(s)ds.Thevalues of β, u0,K,ζ0, c, γA, γB, γD,aredeterminedbythesystemoftenequations:)

ϕ(γB ) − ϕ(2π)=∆ϕTB, ϕ(γA ) − ϕ(γB )=∆ϕBA ϕ(γ ) − ϕ(γ )=∆ϕ , ϕ(0) − ϕ(γ )=∆ϕ " D " A AD " " D DT % % % % ϕ (0)" = ϕ (γD")=ϕ (γA )=ϕ (γB ")=ψ(γ")=ϕ(γA )=0 (16) " " " " " " As a consequence, the function ϕ(γ)isknown.Sinceϕ(γ)=ϕ(s), the function % s = s(γ)isknowninimplicitform.Hencethederivatives (γ)=−|dz/dζ||ζ|=1 is also known, and therefore we" obtain the same problem" of the previous sec- tion, namely that of determining log(dz/dζ)onceitsrealpartisknownonthe contour. Now log(dz/dζ)hastwologarithmicsingularitiesduetothetwoedges A and T .Theseareavoidedbyconsideringthefunction

dz 1 ζA ω(ζ)=P (ζ)+iQ(ζ)=ln −(ε −1) ln 1 − −(ε −1) ln 1 − (17) dζ T # ζ $ A # ζ $ where εT π and εA π are the external angle of the T and A edges, respectively. P (γ)isknownandtheSchwarzformula(10)determinesω(ζ)thatyields the inverse problem solution. Following the same remarks of the simple airfoil case, three constraints on vp(s)enforcingtheclosureconditionsandthecompatibilityatinfinity: 2π 2π 2π P (τ)dτ = B1, P (τ)cosτ dτ = B2, P (τ)sinτ dτ = B3. (18) !0 !0 !0 with u B =2π ln 0 ,B= π ε − 1+(ε − 1) cos γ ,B= π (ε − 1) sin γ . 1 v 2 T A A 3 A A ∞ * + So far the discussion of vortex equilibrium has been neglected. It is quite remarkable that vortex equilibrium implies another two constraints on vp(s) that can be expressed in a form analogous to eq. (18). According to the Routh rule [12], a vortex is standing in equilibrium in the physical z-plane if the velocity ζ˙0 of its free image on the transformed ζ-plane is

˙∗ K d dz ζ0 = − ln (19) 4πi ,dζ #dζ $-ζ0 with iβ ˙∗ −iβ u0e K 1 Γ+K ζ0 = u0e − 2 + − (20) ζ0 2πi ζ0 − ζ0 2πiζ0 According to the Schwarz formula (10) d dz 1 2π 2 eiτ ln = − P (τ) iτ 2 dτ+ ,dζ #dζ $-ζ0 2π !0 (e − ζ0) 1 1 +(εT − 1) +(εA − 1) + (21) ζ0(ζ0 − 1) ζ0 ζ0 − 1 #ζA $ and eq. (19) can be rewritten as 2π 2 cos τ + ρ0 cos(τ − 2γ0) − 2ρ0 cos γ0 P (τ) 2 2 dτ = B4  !0 [ρ0 +1− 2ρ0 cos(τ − γ0)]  (22)  2π 2  sin τ − ρ0 sin(τ − 2γ0) − 2ρ0 sin γ0 P (τ) 2 2 dτ = B5 ! [ρ +1− 2ρ0 cos(τ − γ0)]  0 0  with ρ0 = |ζ0|, γ0 =arg(ζ0)and − − ρ0 cos 2γ0 − cos γ0 ρ0 cos(2γ0 γA ) cos γ0 B4 π ε − ε − = ( T 1) 2 − +(A 1) 2 − − + 2 ρ0(ρ0 +1 2ρ0 cos γ0) ρ0[ρ0 +1 2ρ0 cos(γ0 γA )]

− u0 1 − − 2ρ0 − 2 Γ 4π sin β + 2 sin(β 2γ0) 2 cos γ0 1+ cos γ0 K , ρ0 - ρ0 − 1 ρ0 # K $ 3 − − ρ0 sin 2γ0 − sin γ0 ρ0 sin(2γ0 γA ) sin γ0 B5 π ε − ε − = ( T 1) 2 − +(A 1) 2 − − + 2 ρ0(ρ0 +1 2ρ0 cos γ0) ρ0[ρ0 +1 2ρ0 cos(γ0 γA )]

− u0 − 1 − 2ρ0 − 2 Γ 4π cos β 2 cos(β 2γ0) + 2 sin γ0 1+ sin γ0 K , ρ0 - ρ0 − 1 ρ0 # K $ 3 The five constraints (18),(22) that have to be satisfied in order to obtain a well posed problem have the form

2π P (τ) fk(τ)dτ = Bk con k =1, 2, 3, 4, 5(23) !0 where the functions fk(τ)canbeinferredfromeqs.(18),(22) The constrains are, in general, not satisfied by the P (γ)pertinenttoan arbitrarily given vp(s). Therefore, we look for a quasi solution for a P∗(γ)that satisfies the constraints and is the closest, in the L2 sense, to the original P (γ), that is, we look for the minimum of the functional

2π 2 J(T )= T (τ)f0(τ)dτ !0 where f0(τ)isaweightfunctionandT (τ)=P∗(τ) − P (τ). The minimum is found under the five constraints 2π T (τ) fk(τ)dτ = Ck (24) !0 with 2π Ck = Bk − P (τ) fk(τ)dτ, k =1, 2, 3, 4, 5 !0 According to the Lagrangian multipliers technique, T (τ)isobtainedbyfinding the minimum of the unconstrained functional

5 2π L(T )=J(T ) − µk T (τ) fk(τ)dτ − Ck ,! - 4k=1 0 where µk are the Lagrangian multipliers. Letting the first variation of L(T ), with respect to T (τ), be zero, one obtains

1 5 T (τ)= µkfk(τ)(25) 2f0(τ) k4=1 which, enforcing the constraints (24), yields

5 2π fj(τ)fk(τ) µk dτ = Cj,j=1, 2, 3, 4, 5 ! 2f0(τ) k4=1 0 that allows the computation of the multipliers µk and hence T (τ). The case of a vortex trapping airfoil obtained by solving the inverse problem and satisfying all of the five constraints including the equilibrium conditions for the vortex is shown in figs. 4,5 as an example

0.5 0.4 0.3 y 0.2 0.1

-2 -1.5 -1 -0.5 0 x

Figure 4: Vortex trapping airfoil.

2

1

0

-1

-2

v(s/L) -3

v* (s/L) -4

0.2 0.4 0.6 0.8 1

Figure 5: Original velocity distribution and that one that minimizes the Lagrangian. BIBLIOGRAPHY

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