A comparison of velocity and potential based boundary element methods for the analysis of steady 2D flow around foils

Transcription

1 A comparison of velocity and potential based boundary element methods for the analysis of steady 2D flow around foils G.B. Vaz, L. E a, J.A.C. Falcao de Campos Department of Mechanical Engineering, Institute Superior Tecnico, Portugal Abstract The potential based BEM Morino variant is compared with a modified version of the velocity based method proposed by Hess, on the calculation of potential flow around 2-dimensional foils. In the Morino method, several numerical implementations of the Kutta condition are investigated. The test cases for the comparison of the two methods are analytical foils obtained by conformal mapping techniques: Joukowski and Karman-Trefftz. Surface solutions are presented and the main drawbacks and advantages for each formulation are discussed. A verification study performed for the two test cases reveals the existence of difficulties for the Morino method in foils with a cusped trailing edge. 1 Introduction In most of the original panel methods, see Hess [1], for solving the incompressible potential flow problem for non-lifting and lifting bodies, an integral equation formulation is used in combination with a Neumann boundary condition. This boundary condition requires that the normal velocity must be zero on the body surface. Methods based on this formulation have been classified as velocity based boundary elements methods. Subsequently, Morino et al [2], introduced a method based on Green's formula for the potential, combined with an internal Dirichlet boundary condition on the body surface. Methods based on this formulation have been known as potential based boundary elements methods. Both velocity and potential based methods are in current use and each type of method has its advantages

2 284 Boundary Elements XXII and disadvantages. In this work a comparison of these two approaches for the computation of the potential flow around two-dimensional lifting foils is presented. For the Morino potential based method, the popular low-order implementation is used with constant distributions of dipoles and sources on each panel surface. For the velocity based method, a modification to the usual method by Hess is made. It is used constant sources distributions on each panel surface and a piecewise constant distribution of vortices on the mean line inside the foil, as described in [3]. Problems involving lift require the imposition of the Kutta condition at the trailing edge. In the Morino method, five different numerical implementations of the Kutta condition are tested and their performance evaluated when the methods are applied to foils with cusped trailing edges. Foils with thin or cusped trailing edges are known to be problematic for some BEM, causing ill-conditioning or loss of accuracy [4], [5]. In an attempt to understand the nature of these problems a verification study is performed for a cusped Joukowski foil and a Karman-Trefftz foil with a finite trailing edge angle, and the extrapolated results for a infinite number of elements compared with the analytical solution. 2 BEM Formulation 2.1 Theoretical formulation Consider a closed two-dimensional domain V with boundary S, the unit normal vector n to S being oriented into V, as pictured in Fig. 1. The boundary S is composed of the body surface SB, the wake surface <$w, and the outer control surface S^Q surrounding the body and wake surface. Figure 1: Notation and geometry domain of the problem.

3 Boundary Elements XXII 285 The flow of interest is in the outer region V, where the flow is considered to be incompressible and ir rotational. We consider a uniform onset flow with velocity T/^o, and introduce a potential function $ = ^^ + </>, such that V = V< is the flow velocity, 0oo is the potential of the undisturbed flow, VQO V0oo, and 0 is the perturbation potential due to the presence of the body. The perturbation potential satisfies the Laplace equation: V^ = 0. (1) In a body reference frame, the kinematic boundary condition on the surface writes ^ = 0 # ^ = - V0oo fl = -tl 7%. (2) cm cm In the lifting problem we allow for a discontinuity of the potential across Sw- For the steady lifting problem, the potential jump across the wake surface, is identical to the circulation around the body, and is constant on 5>v: W)on ^ = ^ - ^ = T. (3) As mentioned before, a Kutta condition is required at the trailing edge to uniquely specify the circulation. In its most general form it requires the flow velocity at the trailing edge to remain bounded, i.e. V0 ^ < oo. On the outer control surface S^o it is required that the flow disturbance, due to the body's motion through the fluid, should vanish in the limit where this surface tends to the infinity: V0-4- 0, as J>oo > oo. (4) Applying Green's identity, see for instance Batchelor [6], the potential at a point p is, = - s L ['* <" - ' "» <** "' - where 0' is the potential in the region V interior to the body and we have made use of Eq. (4) and the fact that the normal derivatives of the potential are continuous across 5w- This last equation can be regarded as a representation of the velocity potential in terms of a normal dipole distribution, of strength /z, on SB and <5>v, and a source distribution, of strength a, on SB- For the case of the field point p belonging to the body surface SB, in (Eq. 6), the term is (5)

4 286 Boundary Elements XXII changed to ^. Therefore, for a point on the surface, Eq. (5) can be written as, The Morino variant, known as the perturbation potential method, chooses for the inner potential (f)' (q) the value zero, so that the dipole strength p, = <j) (q) and a -j = Voo ' n is known from the boundary condition (2). For the velocity based formulation, the modified Hess method assumes a distribution of vortices, g (t) on the foil mean line rather than on the surface, and following Ega & Falcao de Campos [3], the integral equation can be stated as: ^_?(P), 1 f,\ 2 2^ / <7(9)i^(log#)&S' H Tr ^ ^ ^ Both methods lead to Fredholm's integral equation of the second kind. For the potential based method, Eq. (6), with the source strength known from the boundary condition (2), is an integral equation on the unknown dipole distribution on the surface. Eq. (7) for the velocity based method, with the left hand side known from the boundary condition (2) is an integral equation on the unknown source distribution on the surface. The Kutta condition is required to determine the potential jump A0 on J>yy in Eq. (6), or the circulation F around the foil in Eq. (7). 2.2 Numerical implementation In the numerical implementation of the Morino method some assumptions must be made in terms of geometry and singularity discretizations. The Morino method applied in this work is low order, with constants distributions of singularities and flat panels. The surface is therefore discretized using a given number, NP, offlatpanels. The middle panel points are chosen as collocation points where the discrete set of equations is satisfied. The stretching used for surface discretization is the full cosine, providing refined zones at the trailing and leading edge. The five different types of Kutta conditions for Morino's method are based on different principles: Pure numerical: simple (Kl) and linear extrapolation (K2) Kutta conditions. The simple or usual Kutta condition requires the potential jump in the wake to be equal to the difference of potential values

5 Boundary Elements XXII 287 of the upper part of the wake and the lower part of wake, Eq. (3). However, this condition is numerically implemented on the last collocation point, not at the real trailing edge. Therefore, applying a linear extrapolation procedure could improve the implementation, Vaz [7]. Analytical potential solutions based: wedge Kutta condition, (K3). In this case the local trailing edge potential is assumed to approach the analytical corner flow solution, Lee [8]. # Combinations of the above: subdivided wedge Kutta condition, (K4). The same as the previous but using local refinement on the trailing edge zone, Lee [8]. Physical constrains based: Non-linear iterative pressure Kutta condition, (K5). Instead of specifying the value of the potential jump on the trailing edge, it is required a zero pressure jump. This condition is the 2-dimensional analogue of the three dimensional non-linear pressure Kutta condition, Kerwin et al [9]. Surface velocities are calculated by means of a second order differentiation scheme of jj, relative to foil arclength, and the pressure coefficient Cp, using Bernoulli's equation can be defined as Cp = 1 - (^7 J. For the modified Hess method we use a piecewise constant vortex distribution along the mean line of the foil to generate the circulation required to satisfy the Kutta condition. The vortex strength on each mean line panel is obtained from the value given by, 7 = 7o3m\ (8) where Sm is the distance to the trailing edge measured along the mean line panels, 70 is a constant to be determined by the Kutta condition, which in this case imposes equal tangential velocity at the collocation points near the trailing edge, Ega & Falcao de Campos [5]. The analytical foils and corresponding solutions for potential, pressure and velocity are obtained by the conformal mapping theory, Vaz [4]. 3 Results and Discussion 3.1 Comparisons of the Methods The basic foil chosen to test the two methods is the Joukowski foil with a maximum thickness and camber to chord ratios of tm/c 0.04 and fm/c 0.02, respectively. It permits the calculation of analytical quantities as potential, 0, velocity V and pressure distribution, Cp. The test angle of attack is chosen as 1.5, since for this kind of foil, already causes a suction peak, and, therefore, harder conditions for the numerical methods.

6 288 Boundary Elements XXII Figure 2: 1/2 norm for <p versus the number of discretization panels NP. An 0 He A Mo V Mo q * Mo Mo ytic no Kl nok2 nok3 no K4 NTE= x/c [%] Figure 3: Pressure distribution on the foil. For the Morino method, after solving the numerical discretized form of Eq. (6) by means of a LU decomposition solver, a study to see the adequate number of panels for a discretization is madefirstlyby means of the primary solution cj>. Fig. 2 shows the Euclidean norm of 0 error versus the number of panels NP. The errors decay with the increase of the number of panels and a NP 160 shows errors less than 0.01 for all the Kutta conditions. The effect of the K2, K3 and K4 conditions on the solution is to decrease the error when the number of panels used is small. However, for high discretization levels K5 is more accurate. Choosing 160 panels as an 'optimum' discretization level in terms of computational effort versus error, Fig. 3 shows the Cp distribution for Morino's and Hess method. It is clear that for this foil thickness, angle of attack and discretization, the calculated pressure distributions show good

7 Boundary Elements XXII agreement with the analytical solution all over the foil f Hess Morino 7 Morino K3»3 Morino K4 NTE=451 Morino K = x/c [%] Figure 4: Absolute error distribution for the pressure distribution coefficient on the foil. Negative x coordinates means foil lowerside. t/c [%] Figure 5: Cip relative error distribution versus thickness, a = 1.5. NP=160. Fig. 4 plots the error in the pressure distribution to more closely show the differences for both methods and variants. Approximately for all kinds of Kutta conditions, the Morino method shows better accuracy near the trailing edge and zones adjacent to the leading edge. Nevertheless, the maximum error, located at the leading edge suction zone, is much lower for the Hess method. The best performance is achieved for K2 and K3. A global measure of the accuracy of the two methods can be obtained from the integral coefficients values calculated by means of pressure integration. For the lift coefficient Cip, which is a quantity of interest, the Hess method gives the most accurate prediction for the same angle of attack and

8 290 Boundai*v Elements XXII thickness. In spite of this, Fig. 5 shows that this situation can be changed for different foil thickness. In the low range of thickness, [0.5%,3%], the Morino method produces better results than the Hess method. For higher thickness the situation is inverted. The pressure iterative Kutta condition, K5, produces less accurate results in the full range of thickness within the conditions of the tests made x/c [%] Figure 6: Pressure distribution on the foil. NP=320. The choice of an 'optimum' discretization level based on Figs. 2 and 3, can be misleading if the convergence of a particular method is to be assessed. As a matter of fact there is a loss of accuracy for some variants of Morino's method with the increase of the number of panels for this foil. This is visible in the pressure distribution opening near the trailing edge, for 320 panels, depicted in Fig. 6. This phenomenon was already reported by Katz [4] and Kinnas [5] for foils with cusped and thin trailing edges. This last fact motivated the verification tests of the methods presented in next section. Table 1: Grid convergence results for Case Hess Morino Kl Morino K2 Morino K3 Morino K4 Morino K5 P Joukowski Q* IQ, - Qe P with Joukowski foil. Karman-Trefftz Clo IQo -QJ Verification Study The main purpose of a verification study is to determine if we are solving the equations right, Roache [10]. A verification study enables the determination

9 Boundarv Elements XXII 291 of the apparent order of accuracy of a method, p, and the extrapolation of the results of a numerical solution to the grid of cell size zero, i.e. in a BEM method to an infinite number of elements. The determination of the apparent order of accuracy of a method, p, is based on the assumption that the error of the numerical solution behaves asymptotically as, error= const x /i?, where hi is the representative panel size. 0.26, h./h, Figure 7: Least square rootfitcurve for C/p Hess and Morino's method (left) grid convergence results and Kutta condition influence on results (right). A validation process certifies if we are solving the right equations, Roache [10], and may be carried out by comparing the numerical solution with experimental results. In the present paper, we present verification studies for two test cases : a cambered Joukowski foil at an angle of attack of 0 and a Karman-Trefftz foil, already used in Lee [8], at an angle of attack of 2.5. The Joukowski foil has a cusped trailing edge, whereas the Karman-Trefftz has a relatively large trailing edge angle of 27. The verification procedure is applied to the lift coefficient, Cip, obtained by integration of the pressure distribution. Eleven geometrically similar discretizations have been considered, with afinestdiscretization of 320 panels and a coarsest of 40 panels. The apparent order of accuracy, p, and the extrapolated value of the lift coefficient, Qo, are obtained from a least square root fit to thefivefinestgrids, which have 320, 280, 240, 200 and 160 panels (The details of the verification procedure are given and discussed in Ega & Hoekstra [11]). Tab. 1 presents the values of p, C% and the difference between C^ and the analytical solution, C%, for the two test cases. The results of the Joukowski foil are shown in Fig. 7(left), and Fig. 7(right) illustrates the effect of the Kutta condition on the results. In all the cases the data follows the expected behaviour, with an excellent agreement between the fitted curves, solid lines, and the data, even for more solutions than the five finest grids.

10 292 For the Joukowski foil, the extrapolated values of the lift coefficient, C^, in the Morino method are not in agreement with the analytical solution. The results suggest that the difference between C^ and Cie are clearly larger than the uncertainty in the determination of Cio- Furthermore, the numerical implementation of the Kutta condition has a significant effect on the apparent order of accuracy, p, and in the extrapolated value of 0.33p 0.31' Figure 8: Least square root fit curve for for a Karman-Trefftz foil. the lift coefficient, C%. On the other hand, the value of Ci<> obtained with the Hess method is in excellent agreement with the analytical value. In the results presented for the Karman-Trefftz foil, Fig. 8, both methods exhibit values of (% in excellent agreement with the analytical solution. The differences between C^ and C% are smaller than 10"^, which is a perfectly acceptable value considering the uncertainty of the estimation of C\^. In Morino's method, the differences between the results obtained with the five numerical implementations of the Kutta condition are much smaller than in the Joukowski foil. The apparent order of accuracy is 1.0 in all the cases, with the exception of the K5 condition which has a p of 1.2. The results obtained with the two methods show that the Morino method is more sensitive to the trailing edge angle than the present version of the Hess method. 4 Concluding remarks The comparison of the potential and velocity based BEMs presented, shows that for levels of discretization accepted in practice, both methods are able to predict with reasonable accuracy the pressure distribution on a Joukowski foil. For small foil thickness, the Morino method shows better results than the Hess method while the opposite occurs for large foil thickness. However, when increasing the number of panels for the Joukowski foil, the Morino method shows increasing errors at the trailing edge. This shows that special care has to be taken before a certain level of discretization is accepted. The verification study performed on the same Joukowski foil reveals that, for the Morino method, the extrapolated value of the lift coefficient

11 Boundary Elements XXII 293 for infinite number of panels differs from the analytical solution. These extrapolated values are influenced by the choice of the Kutta condition. However, for the Hess method, the numerical solution appears to converge to the analytical solution. For the Karman-Trefftz foil the extrapolated values for both methods are in excellent agreement with the analytical solution. This suggests that the present formulation of the Morino method may not be convergent to the analytical solution for foils with cusped or thin trailing edges. This problem deserves further investigation. Acknowledgments The first author acknowledges thefinancialsupport granted by Fundaqao para a Ciencia e a Tecnologia, Ph.D. grant PRAXIS.XXI/BD/22269/9. This work was done under the project PRAXIS/2/2.l/MAR/1723/95. References [1] Hess, J.L. Panel methods in computational fluid dynamics. Annual Review Fluid Mechanics, 22, pp [2] Morino, L. & Kuo, C.C. Subsonic potential aerodynamics for complex configurations: a general theory. AIAA Journal, 12(2), pp [3] Ega, L. & Falcao de Campos, J. Analysis of two-dimensional foils using a viscous-inviscid interaction method. Int. Shipbuild, 40(422), pp [4] Katz, J., Yon, S. & Plotkin, A. Effect of airfoil trailing edge thickness on the numerical solution of panel methods based on the Dirichlet boundary condition. AIAA Journal, 30(2), pp [5] Kinnas, S.A. & Hsin, C.Y. The local error of a low-order boundary element method at the trailing edge of a hydrofoil and its effect on the global solution. Computer & Fluids, 23(1), pp [6] Batchelor, G. An Introduction to Fluid Dynamics. Cambrigde University Press [7] Vaz, G.B. Two-dimensional boundary element application. Morino's method. fgt-m^#et#c-#t-jj<%-2 Tec/tmW Report. April [8] Lee, J.T. A surface panel method for the analysis of hydrofoils with emphasis on local flows around leading and trailing edges. Proceedings of Seminar on Ship Hydrodynamics. S.N.U., May [9] Kerwin, J.E., Kinnas, S.A. et al. A surface panel method for the hydrodynamic analysis of ducted propellers. SNA ME transactions, 95, pp [10] Roache, P. J. Verification and Validation in Computational Science and Engineering, Hermosa publishers: Albuquerque, USA, [11] Ega, L. & Hoekstra, M. On the numerical verification of Ship Stern Flow Calculations. Proc. o/^5^mv4^atet-cfdifora;^op. Barcelona, 1999.