An alternative approach to integral equation method based on Treftz solution for inviscid incompressible flow Antonio C. Mendes, Jose C. Pascoa Universidade da Beira Interior, Laboratory of Fluid Mechanics, Covilhd 6200, Portugal Jan A. Kolodziej Poznan University of Technology, Institute ofapplied Mechanics, 60-965 Poznan, Poland Abstract The paper deals with the numerical modelling of inviscid, incompressible flow about two-dimensional lifting surfaces at low speed. The problem is formulated in terms of the Laplace equation. For its solution two different approaches are discussed. The first one tackles the problem in terms of the velocity potential with normal derivative prescribed at the boundary. An integral equation method based on a distribution of vortices is utilised to solve the flow. As for the second approach the governing equation is written in terms of the stream function. Treftz method is then used to express the solution as a power series of the fundamental solutions of Laplace equation. 1 Introduction Modelling of inviscid incompressible flow about lifting surfaces of an arbitrary geometry may be conducted by means of integral or differential
58 Advances in Fluid Mechanics II representations. The problem is governed by a field equation which has originally been solved by numerical methods applied to a grid extended along the fluid domain. By making use of Green's theorem the solution of the problem may be reduced to solve an integral equation written for the physical boundaries of the flow. In this context the classical approach by surface singularity distributions has originated the well-known panel methods in two and three-dimensions - Hess [1]. Another classical approach, which is not so commonly applied to solve this class of problems, is the Treftz Method - Kolodziej [2], Herein we introduce this methodology as an alternative to the more traditional integral equation approach. In this last approach the problem is formulated in terms of the velocity potential with normal derivative prescribed at the boundary. This harmonic function is afterwards searched for as the velocity field generated by a distribution of vortices along the body contour. The resulting Fredholm integral equation, of the first kind, is subsequently solved for the tangential velocity at a discrete number of panels that are distributed along the body profile. The numerical results that were obtained by modelling a NACA 0012 airfoil have been subsequently compared to those experimentally obtained by the authors in a low-speed wind tunnel. 2 Potential flow equations Let us consider a two-dimensional lifting surface of an arbitrary shaped cross-section, like the one represented in Fig. 1. Its contour ( C ) is a closed line having continuous derivative everywhere except at the trailing edge, whose outward normal n points to the unbounded fluid domain (D). Here OXYZ is a stationary co-ordinate system and oxyz is a system of axes attached to the body. Fig. 1 Wing section moving with speed U and angle of attack a in fluid domain (D).
Advances in Fluid Mechanics II 59 The solid body advances with velocity U in air which is initially at rest; a is the angle between the chord c and the direction of motion of the wing. At low speed the flow induced in the vicinity of the body may be assumed as incompressible. Moreover, if we assume inviscid, irrotational flow, the fluid velocity V may be derived from an absolute potential. This function is the solution of the following system of equations: V^0(z) = 0, (zed) (1) - " ' m where z stands for the complex co-ordinate z=x+iy. The velocity potential is, therefore, the solution of Laplace equation (1) and the impermeability condition (2), a condition to be applied on the solid boundaries of the flow. At infinity a regularity condition applies which may be written in terms of the relative potential: = -U, (z-»±oo) (3) 3 Integral-equation approach The harmonic function formulated above is herein searched for as the real part of a complex velocity potential F(z). This velocity field will be generated in every point z of the fluid domain (D) by a distribution of Rankine vortices with density y(z') at a generic point along the contour (z' At this stage the velocity field associated with this function satisfies automatically to the governing equation (1) and must equally verify the boundary conditions stated above. As to condition (2) imposed on the surface of the body, this condition may be formulated in the integral form as follows:
60 Advances in Fluid Mechanics II = 0.5 (5) taking into account eq (3). In the previous equation the kernel of the integral may be easily determined after differentiation of the function defined in eq (4). The angle 0 represents here the argument of the unit-vector tangent to the contour of the body. Equation (5) is a Fredholm integral equation, of the first kind, which is to be solved in order to the vortex densities y, for a prescribed body motion with velocity U. Fig. 2 Discretized contour of the wing section. The solution of this equation for a body having a contour of arbitrary geometry requires the use of a numerical approach. A numerical treatment that is commonly adopted consists of sub-dividing the contour ( C ) in N linear elements Sj (j=l..,n), as it is shown in Fig. 2. Each of these elements Sj is the support of a uniform distribution of vortices with density yj. Sj is defined by its extremity points Zj and Zj+i, and its argument 8,; Z=z/c are non-dimensional complex co-ordinates. By enforcing the impermeability condition on N control points TJ taken as the middle point of each element of the contour, eq (5) assumes the discretized form:, = M,,U = i..,n) (6) The influence coefficients of the velocity By may be calculated after integration of the left-hand side of eq (5), taking yj constant along each linear boundary element.
Advances in Fluid Mechanics II 61 B,, =0 (i- j); As to the right-hand side of the linear system, n, is the scalar product of the unit-vector in the direction of the body motion to the normal unitvector at the middle point ij=0.5(zi+zi+i). The system of N equations (6) is to be solved for the N unknown normalised densities (yj/u). In order to get a finite value for the velocity at the wing trailing edge and the system to be determined it is, however, necessary to impose an additional condition at this point. Although Kutta condition may be imposed in few different ways, a convenient form for non-cusped airfoils is to ensure equal velocities on either side of the trailing edge, as suggested by Anderson [3]: yi+y#=0. This is achieved by simply replacing the last equation of system (6) by the former relationship. Finally the pressure coefficient is derived from Bernoulli's law: Following Katz & Plotkin [4], the tangential velocity within the fluid adjacent to the profile may be obtained directly from the vortices density that were calculated above, on each panel of the body. The lift coefficient of the airfoil may then be derived from Joukowski theorem as a function of the circulation F: C - F ^L- (9) 4 Treftz solution The governing equation is here written in terms of the stream function Y of the flow and the value of this function is prescribed at the boundary. In two-dimensional potential flow, Cauchy-Riemann relationships applied toeq(l) yell the following Laplace equation:
62 Advances in Fluid Mechanics II (10) The solution for this flow field is now assumed to be the superposition of a uniform stream of constant velocity U and incidence a, with a perturbation of the flow which is due to the presence of the body. k=\ i=\ (11) where A^ are unknown weighting coefficients of the truncated series that may be determined by means of a simple collocation procedure - Collatz [5] - or by using a Galerkin approximation; y\(x,y,xk,yd are trial functions that exactly fulfil eq (10). These fundamental solutions are sources forz = l, vortices for i=2, doublets for i=3 and multipoles for (12) where n -, X X i The co-ordinates (x^.y^) represent here the positions of these singularities placed inside the contour of the airfoil. Solution (11) fulfils exactly the governing equation (10) and must also satisfy the boundary condition required on the surface of the airfoil, where the stream function must be zero: (* \ ^-0). Keeping this in mind, normal and tangential velocities may be obtained after differentiation of equation (11),
Advances in Fluid Mechanics II 63 03) k = \ /=! (14) Here cosg and sing are direction cosines of the unit-vector tangent to the contour of the body, and: g\ 82=- X XL, 3 = X XL, i -4 -t, 5^, (15) /2 ~ ~ /3 = /, = -(z - 2)!r^ '^ cos[(/ - i = 4, 5,... In the numerical procedure used to determine the NxM coefficients A^ in eq (11), the boundary conditions which are to be imposed on the stream function and normal velocity will be fulfilled only approximately, at a finite number of collocation points along the body profile. In this way we are led to an over-determined system of equations
64 Advances in Fluid Mechanics II which must be solved by the least-squares method. Collocation of the stream function and Neumann conditions for (P-l) control points Tp(Xp,yp) defined along the boundary of the airfoil, yell the following system of equations: TV k = \ 7 = 1 AT M (16) Moreover, Kutta condition states that the backward stagnation point coincides with the trailing edge: TV k=\ 7=1 k=\ i=\ (18) k = \ 7 = 1 Two different collocation procedures, both in least squares sense, have been used. In the first one only the stream function condition is imposed along with Kutta condition; (P+2) linear equations are to be solved in this case. As for the second one, the collocation involves stream function and its tangential derivative (normal velocity), and Kutta condition is enforced as well. Equations (16), (17) and (18) constitute a linear system of (2P+1) equations to be solved for NxM A^j unknowns. Eq (14) is then used to calculate Cp from eq (8) and eq (9) delivers the lift coefficient.
Advances in Fluid Mechanics II 65 5 Numerical results and discussion The numerical results presented here have been obtained for a standard lifting surface (NACA 0012). Fig. 3 gives the pressure coefficient along the wing section, calculated by the integral equation method. The values of Cp are compared with the ones obtained by Moran [6] for the same profile. The lift coefficient is shown in Fig. 4 for different angles of attack a. The computational results are here compared to the results of Abbott and Doenhoff [7] for a Reynolds number equal to 6x10, and to our own experiments at Re =2.5x10. We have used for that purpose a low-speed wind tunnel Flint TE44 (0.45x0.45x0.572m), in which the velocity of air can be continuously varied till 30 m/sec. The model is mounted on an electronic three-component balance TE81/E, equipped with piezo-resistive load cells. A preliminary analysis of the uncertainty of the measured forces points out to a precision of 1% during the calibration procedure. The data acquisition is controlled by an A/D Lab PC card from National Instruments working in Lab VIEW environment. The signals are afterwards processed in order to evaluate the lift and drag coefficients, as well as the aerodynamic moment. The speed of the approaching stream is controlled at the main section of the tunnel by a Prandtl probe. 5-4 -3 Cp -2-1 0 1 0 1 0.2 0.4 0.6 0.8 x/c Fig. 3 Pressure coefficient along the upper and lower surfaces of the NACA 0012 airfoil. For an angle of attack till about 5 we have obtained a good correlation between the numerical predictions and the experimental results. For oc=10 the lift coefficient as predicted by the integral equation model, differs from the experimental value by about 25%. However, this fact is due to the separation of the flow and the
66 Advances in Fluid Mechanics II development of a wake behind the wing. Results comparable to the ones provided above are attained by using the concept of boundary collocation with adaptation - Golik & Kolodziej [8]. Fig. 4 Lift coefficient for the NACA 0012 wing section as a function of the angle of attack. The previous results were calculated for an average of N=200 panels distributed over the body surface. Fig. 5 shows the evolution of the values obtained for CL as we refine the discretization of the boundary. For the NACA 0012, a=8.3 is a particular angle of attack for which CL=!. By applying Joukowski theorem we have obtained a lift coefficient 2% close to this value (0^=0.98), using an average of N=100 panels. From direct integration of the pressure around the wing a more rough estimate of this coefficient is obtained (C[j=0.93), and this is certainly due to the numerical oscillations on the tangential velocity near the trailing edge. The effect of these unwanted oscillations on pressure at the vicinity of this point is visible in Fig. 3. Under these theoretical conditions the wing experiences a small drag Co=0.15, that can be seen as a residual error in fulfilling Kutta condition at the trailing edge of the wing section. This numerical difficulty may be overcome by reasoning on the field induced inside the body - Katz & Plotkin [4] - or by considering higher-order singularity distributions. In general we may conclude that the Integral Equation method reproduces correctly the flow properties. If vortices are used, a formulation based on the normal velocity leads inevitably to an integral equation of the first kind. Consequently, the matrix of the influence coefficients does not exhibit a dominant diagonal; the number of panels that we need to input in order to get an accurate solution is in fact high, if a direct elimination solver with full pivoting is used - Press et al [9]. The
Advances in Fluid Mechanics II 67 improvement gained by using a singular value decomposition had no significance in this case. Condition number based on the Euclidean norm is situated in our case between 10* and 10^, for N=10 to 200 boundary elements. Therefore, double precision is required in order to obtain increased accuracy of the numerical results. As it has been pointed out by Nathman [10] the evaluation of the relevant parameters using higher precision may have some impact on the final results for the case of very thin wings. However, the logarithmic nature of the velocity coefficients in eq (7) will certainly be an important limitation, with respect to the simulation of wings with small thickness. 0 20 40 60 80 100 120 140 N Fig. 5 Convergence of the lift coefficient with number of panels. As to the CPU time surveyed on calculations performed by integral equation method on a PC double Pentium processor at 200MHz, with 64 Mbyte RAM and symmetrical multi-processing, it is of the order of msec for N up to 900 and between 1 and 2 sec for N up to 3000 panels. After this value CPU time starts from 22 sec, as a result of limitations in memory space. Acknowledgements The research work that is presented in this paper was carried out within the framework of PRAXIS grant 3/3.1/CTAE/1940/95. The work reflects a close collaboration between Universidade da Beira Interior (Portugal) and The Technical University of Poznan, in Poland. The numerical and experimental results have been obtained at the Laboratory of Fluid Mechanics of Universidade da Beira Interior.
68 Advances in Fluid Mechanics II References [1] Hess, J. L., Panel methods in computational fluid dynamics, Ann. Rev. Fluid Mechanics, 22, pp. 255-274, 1990. [2] Kolodziej, J. A., Review of application of boundary collocation method in mechanics of continuous media, Solid Mechanics Archives, 12, pp. 187-231, 1987. [3] Anderson, J. D. Jr., Fundamentals of Aerodynamics, McGraw-Hill International, New York, pp.258-263, 1991. [4] Katz, J. & Plotkin, A., Low-Speed Aerodynamics: From Wing Theory to Panel Methods, McGraw Hill, New York, 1991. [5] Collatz, L., The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin, 1966. [6] Moran, J., An Introduction to Theoretical and Computational Aerodynamics, John Wiley & Sons, New York, pp 241, 1984. [7] Abbott, I. & Doenhoff, A., Theory of Wing Sections, Dover Publications, pp 462-463, New York, 1959. [8] Golik W. L. & Kolodziej, J. A., An Adaptive Boundary Collocation Method for Linear PDEs, Numerical Methods for Partial Differential Equations, 11, pp 555-560, 1995. [9] Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T., Numerical Recipes, Cambridge University Press, Cambridge, pp 26-29, 1992. [10] Nathman, J., Precision Requirements for Potential Based Panel Methods, AIAA Journal, Vol. 32, 5, pp. 1089-1090, 1993.