A Kutta Condition Enforcing BEM Technology for Airfoil Aerodynamics

G. lannelli*, C. Grille*, L. Tulumello*

Department of Mechanical and Aerospace Engineering and Eng. Science, University of Tennessee, Knoxville, USA

* Department of Mechanical and Aeronautical Engineering, University of Palermo, Italy

Abstract

An analytical perturbation stream function procedure is established for potential flows about lifting airfoils, which exactly enforces the fundamental

Kutta condition. Using Green's identity, this formulation is transformed into an equivalent boundary integral equation. A boundary finite element discretization of this equation then yields a linear system for the efficient determination of the airfoil surface tangential velocity and associated lift

1 Introduction

For many years boundary integral equation procedures, or panel methods, have been widely used for determining the potential flow field about com- plex aerodynamics configurations*. In the potential function procedure, the potential values are related to the boundary potentials and associated normal derivatives, whereas in the singularity procedure the local potential function value is expressed in terms of boundary sources, doublets and vor- tices of appropriate intensity. In either case, the reliability of the computed results critically depends on accurate boundary distributions of singularities or potentials. Furthermore, the Kutta condition, requiring the aft stagnation point to coincide with the trailing edge, must be adequately enforced. Several implementations of the potential function procedure indirectly en- force the Kutta condition. Conversely, the singularity method employs a trailing edge discretization node where the vanishing velocity condition can be enforced. Nevertheless, it has been reported that the substitution of the trailing edge integral equation with the vanishing velocity condition may lead to solution non-uniqueness.

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In this paper, the potential flow problem is solved in terms of the perturbation stream function ijj. The complete differential statement for ty is derived in the form of a mixed Dirichlet/Neumann boundary value problem that naturally provides for a direct enforcement of the Kutta condition without dispensing with the associated trailing-edge integral equation. The Dirichlet data are the values of ip along the airfoil surface exclusive of the trailing edge, while the Neumann constraints are the values of the normal derivatives of i\) in the neighborhood of the trailing edge. Using Green's identity, the governing differential equation and associated boundary conditions are cast into an equivalent integral equation. The terminal system of discrete linear equations is generated via a boundary finite element formulation that couples the boundary values of the stream function and its normal derivatives. A linear element discretization is employed, placing a discretization node at the airfoil trailing edge. Thereafter, the Kutta condition is directly imposed in the discrete equations by forcing vanishing velocity components at this critical node. The terminal linear system is solved via Gauss elimination, and several comparisons between the numerical results and theoretical and experimental data validate the procedure.

2 Governing Equations

The total stream function T/VOT for an irrotational, inviscid, incompressible steady flow satisfies the Laplace's equation

, ^Tor n • n^%>2 n\ + as = 0 in 0 C %T (1) 2 along with adequate Dirichlet/Neumann boundary conditions. For this equation, fi is the 2-D domain external to the airfoil, see Figure la, bounded by d$l = Ta U TOO, where F& denotes the airfoil surface and F^ indicates the infinity circle. The cartesian components of the velocity vector V and its magnitude V are then determined through

(2/rt)\

The 2-D potential flow problem is recast via introduction of the perturbation stream function ^ defined by

This relation allows deriving an integral statement for ip that both dispenses with the boundary F^ and leads to an accurate perturbation aerodynamic flow fielddetermination . For an arbitrary angle of attack a and free stream velocity %%>, the stream function ijj^> is expressed as

^oo = T4o(?/cosa — xsino;) (4)

Consequently, insertion of (3) into (1) shows that ip also satisfies Laplace's equation

—— H = 0 in H C yt* (5) subject to appropriate boundary conditions.

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3 Far-Field Boaundary Conditions

The far-field condition for ip follows from (3). The limits of (3) and its normal derivative are

lim i/j(x) = 0 , lim = 0 (6)

These relations guarantee for unbounded \x\ the existence of suitable positive numbers TI,

- |r ^ « <- Consequently,

0 < lim (|*|) • ||| < lim n(|%|) . -2_ = 0 (8) - ^' '/

These relations are instrumental for the enforcement of the far-field boundary condition.

4 Wall Tangency Boundary Condition

The fundamental wall tangency boundary condition along the airfoil surface Fa, see Figure Ib, is expressed as V • n = u - HX + v • Uy = 0 (9) where n = {n^.nyY denotes the outward pointing normal unit vector. In the same figure 5 = {s^, sj^ = {^/(fr, (f?//(ir}^ = {-^, ^}^ denotes the tangential unit vector. Consequently, insertion of the velocity components from (2) into (9) yields

9^Tor <%AroT n — o- ' KX -- -- • Uy = 0 => #?/ OZ * d^TOT dy d^Tor dx _ d^or _ , . a% 'dr a% "dr" ar " ^"J

Therefore, as well known, the value of I/JTQT is constant along the airfoil surface, hence ^TOT(^ra) = const (11) Denoting this constant by ^TOT' the trailing edge value of T/VOT, the wall tangency constraint on -0 is then expressed as

^(^rj = VôT - ôo(^rj = ôT - Wz/r* coso; - z^ sina) (12) The specification of I/JÔT ^ addition to the first relation in (6), yields a complete Dirichlet problem specification for ifr and equation (5), which in this case admits a unique solution continuously depending on the boundary data. Consequently, prescribing an arbitrary value for ipôr would yield an analogously arbitrary solution. Thus, i/j™^ cannot be arbitrarily specified. On the contrary it must be obtained as part of the solution itself. There- fore, an alternative, (Neumann), boundary condition must be imposed in the neighbourhood of the trailing edge to isolate the unique physically mean- ingful solution for 0.

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5 Kutta Condition

The Kutta condition requires that the airfoil rearward stagnation point coincides with the trailing edge. Hence, both the normal and tangential velocity must vanish in a suitable neighbourhood of the trailing edge, see Figure Ib. The vanishing trailing-edge tangential velocity constraint is expressed as

V.g-==%.,s^4-%.Sy~ = 0 ; y-s+ = %-s++?;.s+=0 (13) and insertion of the velocity components form (2) into (13) yields

(14) dy

The specification of Neumann data at a corner translates into two constraints on the local normal derivatives of if). Therefore, the fundamental trailing edge Kutta condition is rigorously enforced using (12) in addition to the following two trailing-edge tangential velocity relations obtained from (14)-(15) and (3)-(4)

* cos a-f -sin TE

(16)

6 Integral Statement

The complete boundary value problem for ip

) =0 in

lira V(%) = 0 on

. , , at {XTE}

(17) is recast into an equivalent integral equation via the Green's identity

(is) an on Van on

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Insertion of (5) into this relation yields

dr - dr (19) an dn Van 9n which is reduced to a boundary integral statement via the following speci- fications. The function 0 in (18)-(19) is selected as the Green's function

V^(x,x) = c-6(x-x) (20) In this expression, S(x - x) is the Dirac delta function centred at x, and c is a single constant defined as

c=l<^%En-cXl ; c=^-<^zE(Xl ; c = 0<4>%0n (21) where /3 is the positive angle included in fi by the two boundary tangents at x, see Figure Ic. Insertion of (20) into (19) leads to the terminal boundary integral statement

- , uTl J d£l UTl which supplies the value ^(x) at any point x G Q in terms of the boundary values of T/,(%), 9(7/;(a;))/972, 0(x,z) and 9(^(x^))/(9n. In (22) the vari- able of integration is x, while x remains free. Consequently, expression (22) constitutes a function of x. The far-field boundary condition for ^ is easily enforced in this expression. For the external domain SI, each integral along TOO in (22) vanishes for growing |x|. The first integral vanishes since so do both ip and the normal derivative of (/> according to the first relations in (6) and (20). The second integral also vanishes by virtue of (8). Consequently the integral statement for ^ with embedded far-field boundary conditions is the Fredholm integral equation of the second kind to be evaluated along the airfoil surface r%

(%,;r)A(TXa,)) dF (23) C/7~l

Equation (23) is valid at all points x in the unbounded domain f^UTa, and the unknowns are the values of both ip™^ and the perturbation tangential velocity dip/dn along the airfoil surface, exclusive of the trailing edge. For the completion of the rigorous Kutta condition enforcement, the constraints (16) on the normal derivative of -0 are then directly inserted in this equation.

7 Numerical Procedure

The discretization F^ of F& is accomplished via subdivision of the airfoil surface into "TV" linear elements Fj, sequentially numbered in the clockwise direction, see Figure Id. Hence,

r, ~ r* = (j r, (24)

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The variation of i/> and its normal derivative over each element Fj are ap- proximated via the classical finite element expansion

^(%r.) - ^(77) + ^22(77) i

In (25), ipij, ip2j, (dip/dn)ij and (d^/dn)^- denote nodal values at nodes 1 and 2 in the representative boundary element Fj. The linear shape functions Ni, i=l,2, are defined as

Ni(ri) = \(l-rf) , Nt(ri) = ±(l + ri) - 1 < r? < +1 (26)

These 1-D basis functions allow both a straightforward isoparametric coordinate transformation, from the cartesian coordinates along F^ to the rectilinear coordinate 77, and a direct determination of the line element cfF for the Gauss numerical integration of (23). The transformation is

%r. = %(%,) = %ij7Vi(r7) + 2:2^2(77) (27)

where x\j and x^j stand for the cartesian coordinates of element nodes 1 and 2. Insertion of (25) into (23) yields the fully discrete integral equation

^ I", /• *,/ ^ /.u ~.\ „ , r „, \ 9 fj., ~^\ + \ Z> \lh-Mrli •j / / /V^\\'l i I Tl I) f\ I fn\V\^r-i*L)I T* T* 1 1l //L6 I 1I ^ 7/)S^2n •? / / A/<~^^2\'i> I 7~l I) o Ily'V'^r,?^^ fnI T* T* I I] ~^ [ ypj (772 ^ -^ / yrj Cm ^ ^ /

N (9?/) \ f / 9i/) \ r -E On) Ij^hi, ^ ^'^ (dn\""/2j) Jr' , (28) Exactly 'W discrete equations can be formulated by setting x equal to the i^ airfoil surface node Xi for i = 1, 2, • • •, N. The resultant equation system is recast as

(29)

where subscript "j" denotes an element index, and the coefficients h\j, /i|j, gjj, and g?j follow from inspection of (28). At any boundary node "j" the stream function is continuous, hence

^ = ^2; = V^lU+l) (30) where the value of -0 at node "j" coincides with the value at node two in element "j" and the value at node 1 in element "j -f 1". The normal

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derivative of ip is continuous along smooth airfoil sections , but discontin- uous at the trailing edge. Consequently, the following relations apply in a neighbourhood of node "j"

dn \dn\dn) dn

The first relation holds on a smooth arc, while the second is in general valid, and ought to be employed, at any geometric-discontinuity node. Note that the interpretation of (31) is analogous to that of (30). These relations lead to the definition of the global coefficients Hij and GIj as

(32)

= 2& (33) where Sij is the Kronecher delta, subscript "j", j — 1,7V, now denotes a node index, and node TV coincides with the trailing edge. These resolutions follow from the utilization of linear (or higher order) elements with associated trailing edge node. Importantly, they allow a separate contribution from the two values (dip/dn)^~ and (difr/dri)^^ from (16). Insertion into system (29) of (30)-(33) and the Kutta condition relations (12) and (16) respectively, see (17), yields the terminal linear system

EG, j=i '" \c;rV N-

The Ihs unknowns in this system are the "TV - 1" values of the perturbation stream function normal derivatives and the value ^OT of the total stream function along the airfoil. These are uniquely determined by the "JV" equations (34), since the matrix in this system is not singular.

8 Discussion and Results

The developed procedure has allowed the determination of reliableflo wfield s about several airfoils at various angles of attack a. Total velocity, pressure and lift coefficients are computed and critical comparisons are established with available theoretical and experimental results. For the aerodynamic fieldabou t the NACA 23012 airfoil, documentary numerical results are graphed in Figures 2-3. The distribution of the computed surface velocity for a — 0° and a = 8° in Figure 2 is smooth includ- ing the neighborhood of the trailing edge where velocity monotonically de- creases to the Kutta condition zero value. This smooth variation is reflected in the pressure coefficient distributions. These numerical results virtually

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coincide with the theoretical data. Figure 3 graphs the experimental and computed lift coefficient as determined for -8.0° < a < +16.0°. The numerical predictions coincide with the experimental values for a < 10°. The largest deviations occurs at a c± 10° where incipient separation reduces the validity of potential flow analyses.

Concluding Remarks

An analytical perturbation stream function formulation has been derived for 2-D lifting airfoil potential aerodynamics and cast as an integral equation. A boundaryfinit e element procedure has been thereafter developed and val- idated for numerical solution determinations. The only input requirements are the airfoil surface coordinates in addition to the values of angle of attack and unperturbed flow stream function. Significantly, the fundamental Kutta condition is enforced in a rigorous and unambiguous manner. The described linear element implementation allows usage 01 two normal derivatives per node, hence the insertion into the algebraic system of the two separate trailing-edge values of the normal derivative of the perturbation stream function t/\ Several numerical benchmark tests at various angles of attack have con- sistently supported the validity of the method, since the computed surface velocity, pressure and lift coefficients vitually coincide with the experimental data. These results have been, furthermore, achieved with an efficient "small" number of discretization nodes.

References

[1] L. Morino, K. C. Chang, "Subsonic Potential Aerodynamics for Com- plex Configurations: A General Theory", AIAA J., 24 , 2, 1986

[2] C. Grille, G. lannelli, L. Tulumello, "An Alternative Boundary Ele- ment Method Approach to The 2D Potential Problem Around Air- foils", Eur. J. Mech., B/Fluids, 9 , 6, 527-543, 1990

Figure 1: Geometric Details, a) Computational Domain, b) Airfoil Surface, c) Angle (3 Illustration, d) Airfoil Discretization.

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\l \

M v i

v/v. v/v. M w

Figure 2: NACA 23012 Airfoil, N = 24 Nodes. Theoretical and Computed Velocity, a) a = CT, b) a = 8°. + = Theoretical, D = Computed.

/ / / /

L/ c, -i / / / / / / /

•uf- 41- '

Figure 3: NACA 23012 Airfoil, TV = 24 Nodes. Theoretical, Computed, and Experimental Coefficient of Pressure (a = 8°), and Coefficient of Lift

(-8* < a < 16°). -f = Theoretical, D = Computed, o = Experimental.