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A Kutta Condition Enforcing BEM Technology for Airfoil Aerodynamics

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A Kutta Condition Enforcing BEM Technology for Airfoil Aerodynamics Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 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 po- tential 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 stagna- tion 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. Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 186 Boundary Element Technology XII In this paper, the potential flow problem is solved in terms of the per- turbation stream function ijj. The complete differential statement for ty is derived in the form of a mixed Dirichlet/Neumann boundary value prob- lem 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 con- ditions are cast into an equivalent integral equation. The terminal system of discrete linear equations is generated via a boundary finite element for- mulation 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 con- dition 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 numer- ical 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 pertur- bation 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 field determination. 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. Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Element Technology XII 187 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 posi- tive numbers TI, <Ji, r^ and 0*2 such that - |r ^ « <- Consequently, 0 < lim (|*|) • ||| < lim n(|%|) . -2_ = 0 (8) - ^' '/ These relations are instrumental for the enforcement of the far-field bound- ary 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 = V^oT - ^oo(^rj = ^oT - Wz/r* coso; - z^ sina) (12) The specification of I/J^OT ^ 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 ip^or 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. Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 1 88 Boundary Element Technology XII 5 Kutta Condition The Kutta condition requires that the airfoil rearward stagnation point coin- cides 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 con- straints 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 On . , , at {XTE} (17) is recast into an equivalent integral equation via the Green's identity (is) an on Van on Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Element Technology XII 1 89 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.
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