Hindawi Publishing Corporation Journal of Volume 2014, Article ID 676912, 10 pages http://dx.doi.org/10.1155/2014/676912

Research Article On the Kutta Condition in over

Farzad Mohebbi and Mathieu Sellier

Department of Mechanical Engineering, The University of Canterbury, Private Bag Box 4800, Christchurch 8140, New Zealand Correspondence should be addressed to Farzad Mohebbi; [email protected]

Received 30 October 2013; Revised 17 February 2014; Accepted 17 February 2014; Published 1 April 2014

Academic Editor: Ujjwal K. Saha

Copyright © 2014 F. Mohebbi and M. Sellier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a novel method to implement the Kutta condition in irrotational, inviscid, incompressible flow (potential flow) over an airfoil. In contrast to common practice, this method is not based on the panel method. It is based on a finite difference scheme formulated on a boundary-fitted grid using an O-type elliptic grid generation technique. The proposed algorithm uses a novel and fast procedure to implement the Kutta condition by calculating the stream function over the airfoil surface through the derived expression for the with both finite trailing geed angle and cusped trailing edge. The results obtained show the excellent agreement with the results from analytical and panel methods thereby confirming the accuracy and correctness of the proposed method.

1. Introduction airfoil surface and hence the determination of the pressure coefficients. These methods have been extensively investi- The advent of high speed digital computers has revolution- gated in the aerodynamics literature [2–6], so these will not ized the numerical treatment of problems. be discussed any further here. The interested reader can refer Numerical methods, nowadays, have become a routine tool to the above references for further information. However, to investigate fluid flows over the bodies such as airfoil. dealing with the panels and their attributes is numerically Amongst such fluid flows, incompressible potential flows are much more complex than the method proposed in this paper of crucial importance in studying the low-speed aerodynam- and of high programming effort. The Kutta condition should ics problems. The limitations associated with the exact (ana- be introduced into the computational loop in order to solve lytical) solutions with complex variables methods (conformal the derived system of equations for the panel strengths. mapping) motivated fluid dynamicists to develop numerical In this paper, we propose a novel method to numerically techniques to solve incompressible potential flow problems solve the incompressible potential flow over an airfoil which (the Laplace’s equation) over an airfoil. Since the late 1960s, is exempt from considering the quantities such as the vortex the panel methods have become the standard aerodynamic panel strength and . This method takes advantage tools to numerically treat such flows [1]. Panel methods are applicable to any fluid-dynamic problem governed by of an O-type elliptic grid generation technique to generate Laplace’s equation. In these methods, the airfoil surface is thegridovertheflowdomainandapproximatetheflowfield divided into piecewise straight line segments or panels and quantities such as stream function, velocity, and pressure at singularities such as source, doublet, and vortex of unknown the grid points. The Kutta condition is implemented into strength are distributed on each panel. Panel method used the computational loop by an exact derived expression. An for the simulation of an incompressible potential flow past expression is derived for the finite-angle and cusped trailing an airfoil is concerned with the vortex panel strength and edges.Finally,theobtainedresultsfromtheproposedmethod circulation quantities and the evaluation of such quantities are compared to those from the standard literature (both results in the calculation of the velocity distribution over the analytical and numerical) through several test cases. 2 Journal of Aerodynamics

V∞ a function of velocity only, one can obtain the pressure at any point in the flow region, as will be shown.

2.3. Pressure Coefficient. The pressure coefficient 𝐶𝑝 is defined as 2 𝑝−𝑝∞ 𝑉 𝜓 = constant V∞ 𝐶 ≡ =1−( ) . V∞ 𝑝 2 (4) (1/2) 𝜌∞𝑉∞ 𝑉∞

At standard sea level conditions, 3 5 2 𝜌∞ = 1.23 kg/m ,𝑝∞ = 1.01 × 10 N/m . (5) y

x 3. Grid Generation V∞ We have now presented all relations needed to obtain the Figure 1: Boundary conditions at infinity and on the airfoil surface pressure distribution in an incompressible, irrotational, invis- (no penetration). cid flow over an airfoil. To calculate the pressure at any point in the flow region, a grid should be generated over the region.TheellipticgridgenerationproposedbyThompson 2. Governing Equation for Irrotational, et al. [7] is based on solving a system of elliptic partial Incompressible Flow: Laplace Equation differential equations to distribute nodes in the interior of the physical domain by mapping the irregular physical domain Consider the irrotational, incompressible flow over an airfoil from the 𝑥 and 𝑦 physical plane (Figure 2)ontothe𝜉 and 𝜂 (Figure 1). The flow is governed by the Laplace’s equation computational plane (Figure 3), which is a regular region. It 2 ∇ 𝜓=0(𝜓 isthestreamfunction).Theboundaryconditions is based on solving the Poisson equations as follows: are as shown in Figure 1. 𝜉𝑥𝑥 +𝜉𝑦𝑦 =𝑃(𝜉,𝜂), (6) 2.1. Conditions at Infinity. Far away from the airfoil surface 𝜂 +𝜂 =𝑄(𝜉,𝜂), (toward infinity), in all directions, the flow approaches the 𝑥𝑥 𝑦𝑦 uniform free stream conditions. If the angle of attack (AOA) where 𝜉 and 𝜂 are the computational coordinates correspond- is 𝛼 and the free stream velocity is 𝑉∞, then the components ing to 𝑥 and 𝑦 in the physical coordinate, respectively. 𝑃 and oftheflowvelocitycanbewrittenas 𝑄 are grid control functions which control the density of grids 𝜕𝜓 towards a specified coordinate line or about a specific grid 𝑢= =𝑉 𝛼, 𝜕𝑦 ∞ cos (1) point. To find an explicit relation for 𝑥 and 𝑦 in terms of grid points 𝜉𝑖 (𝑖∈[1,𝑀])and𝜂𝑗 (𝑗∈[1,𝑁]), the following 𝜕𝜓 relations may be used: V =− =𝑉 𝛼, (2) 𝜕𝑥 ∞ sin 𝛼𝑥 −2𝛽𝑥 +𝛾𝑥 =−𝐽2 (𝑃 (𝜉, 𝜂)𝑥 +𝑄(𝜉,𝜂)𝑥 ), 𝑢 V V 𝜉𝜉 𝜉𝜂 𝜂𝜂 𝜉 𝜂 where and are components of velocity vector ;thatis, (7) V =𝑢i + Vj i j 𝑥 𝑦 2 ( and are the unit vectors in and directions, 𝛼𝑦𝜉𝜉 −2𝛽𝑦𝜉𝜂 +𝛾𝑦𝜂𝜂 =−𝐽 (𝑃 (𝜉,𝜉 𝜂)𝑦 +𝑄(𝜉,𝜂)𝑦𝜂), resp.). where 2.2. Condition on the Airfoil Surface. For , flow 𝛼=𝑥2 +𝑦2, cannot penetrate the airfoil surface. Thus the velocity vector 𝜂 𝜂 must be tangent to the surface. This wall boundary condition 𝛽=𝑥𝑥 +𝑦𝑦 , canbeexpressedby 𝜉 𝜂 𝜉 𝜂 (8) 2 2 𝜕𝜓 𝛾=𝑥𝜉 +𝑦𝜉 =0 𝜓= , (3) 𝜕𝑠 or constant 𝐽=𝑥𝜉𝑦𝜂 −𝑥𝜂𝑦𝜉 (Jacobian of transformation) . where 𝑠 is tangent to the surface. In the problem of the flow over an airfoil, if the free stream velocity 𝑉∞ and the angle of The solution of the above equations (using the finite dif- attack 𝛼 are known, from the boundary conditions at infinity ference method to discretize the terms) gives 𝑥 and 𝑦 (see (1)and(2)) and the wall boundary condition (see (3)) coordinates (in the physical domain) of coordinate (𝑖, 𝑗) in one can compute the stream function 𝜓 at any point of the the computational domain. physical domain (flow region). Then, by knowing 𝜓,onecan The O-type elliptic grid generation is employed here compute the velocity of all points in the physical domain. which results in a smooth and orthogonal grid over the airfoil Since, for an incompressible flow, the pressure coefficient is surface. The O-type elliptic grid generation technique has the Journal of Aerodynamics 3

M N +N +3 ( , 1 2 ) N2 (M,N1 +2) 4. Solution Approach D C Since 𝑉∞ and 𝛼 are known, the stream function 𝜓 at any point N=2N1 +2N2 +N3 +6 in the physical domain can be obtained from (1)and(2)as N1 follows:

𝜓𝑖 =𝜓𝑗 +(𝑦𝑖 −𝑦𝑗)𝑉∞ cos 𝛼, A (1, 1) M B (M, 1) (9) N3 H(1, N) M G (M, N) 𝜓𝑖 =𝜓𝑗 −(𝑥𝑖 −𝑥𝑗)𝑉∞ sin 𝛼,

where subscripts 𝑖 and 𝑗 refer to any two arbitrary grid points N1 at the physical domain boundaries. Equations (9)areapplied to vertical and horizontal boundaries of the physical domain, respectively. By knowing the values of stream function 𝜓 on E F y boundaries of the physical domain as well as on the airfoil (M N +N +N +4 N , 1 2 3 ) 2 (M,N1 +2N2 +N3 +5) surface (from wall boundary condition), we can obtain the x values of 𝜓 overthephysicaldomain.Sincewedealwith Figure 2: The physical domain. O-type scheme and discretization Laplace’s equation, it is necessary to find relationships for the of the boundaries. transformation of the first and second derivatives of the field variable 𝜓 with respect to the position variables 𝑥 and 𝑦.By usingthechainrule,itcanbeconcludedthat

𝜕𝜓 𝜕𝜓 𝜕𝜉 𝜕𝜓 𝜕𝜂 𝜕𝜓 𝜕𝜓 = + = 𝜉 + 𝜂 , 𝜕𝑥 𝜕𝜉 𝜕𝑥 𝜕𝜂 𝜕𝑥 𝜕𝜉 𝑥 𝜕𝜂 𝑥 (1, N) G (M, N) H M (10) 𝜕𝜓 𝜕𝜓 𝜕𝜉 𝜕𝜓 𝜕𝜂 𝜕𝜓 𝜕𝜓 F (M,N1 +2N2 +N3 +5) = + = 𝜉 + 𝜂 . 𝜕𝑦 𝜕𝜉 𝜕𝑦 𝜕𝜂 𝜕𝑦 𝜕𝜉 𝑦 𝜕𝜂 𝑦 E (M,N1 +N2 +N3 +4)

N = 2N1 +2N2 +N3 +6 N D (M,N1 +N2 +3) By interchanging 𝑥 and 𝜉,and𝑦 and 𝜂,thefollowing relationships can also be derived: C (M,N1 +2) 𝜕𝜓 𝜕𝜓 𝜕𝑥 𝜕𝜓 𝜕𝑦 𝜕𝜓 𝜕𝜓 𝜂 = + = 𝑥 + 𝑦 , 𝜕𝜉 𝜕𝑥 𝜕𝜉 𝜕𝑦 𝜕𝜉 𝜕𝑥 𝜉 𝜕𝑦 𝜉 (1, 1) A M B (M, 1) 𝜉 (11) 𝜕𝜓 𝜕𝜓 𝜕𝑥 𝜕𝜓 𝜕𝑦 𝜕𝜓 𝜕𝜓 = + = 𝑥 + 𝑦 . Figure 3: The computational domain showing the discretization of 𝜕𝜂 𝜕𝑥 𝜕𝜂 𝜕𝑦 𝜕𝜂 𝜕𝑥 𝜂 𝜕𝑦 𝜂 the physical domain boundaries. By solving (11)for𝜕𝜓/𝜕𝑥 and 𝜕𝜓/𝜕𝑦,wefinallyobtain

𝜕𝜓 1 𝜕𝜓 𝜕𝜓 = (𝑦 −𝑦 ) , 𝜕𝑥 𝐽 𝜂 𝜕𝜉 𝜉 𝜕𝜂 (12) 𝜕𝜓 1 𝜕𝜓 𝜕𝜓 advantage that the grid around the airfoil is orthogonal. The = (−𝑥 +𝑥 ) , discretization of the physical domain and the corresponding 𝜕𝑦 𝐽 𝜂 𝜕𝜉 𝜉 𝜕𝜂 (13) computational domain are shown in Figures 2 and 3,respec- 𝑀 𝑁=2𝑁 +2𝑁 + tively. In the computational domain, and 1 2 where 𝐽=𝑥𝜉𝑦𝜂 −𝑥𝜂𝑦𝜉 is Jacobian of the transformation. To 𝑁 +6 𝜉 𝜂 3 are the number of nodes in the and directions, transform terms in the Laplace equation, the second order respectively. The resulting O-type grid scheme over an airfoil derivatives are needed. Therefore, one has the following. 𝑁 =𝑁 𝑁 =2𝑁 −1 𝑁=6𝑁 +5 for the case 2 1 and 3 1 or 1 is In the physical domain (𝑥, 𝑦), shown in Figure 4. The initial guess for the elliptic grid generation is per- 𝜕2𝜓 𝜕2𝜓 ∇2𝜓= + =0. formed using the Transfinite Interpolation (TFI) method. 𝜕𝑥2 𝜕𝑦2 (14) Since the TFI method is an algebraic technique and does not need much time to generate grids over the physical domain, (𝜉, 𝜂) it will be an appropriate initial guess for the elliptic grid After transformation, in the computational domain , generation method and accelerate the convergence time for 1 the elliptic grid generation method. Another advantage of ∇2𝜓= (𝛼𝜓 −2𝛽𝜓 +𝛾𝜓 ) + [(∇2𝜉) 𝜓 + (∇2𝜂) 𝜓 ] , 𝐽2 𝜉𝜉 𝜉𝜂 𝜂𝜂 𝜉 𝜂 using the TFI method as an initial guess is that it prevents thegridsgeneratedbyelliptic(O-type)methodfromfolding. (15) 4 Journal of Aerodynamics

0.6 0.15

0.4 0.1

0.2 0.05

y 0 y 0

−0.2 −0.05

−0.4 −0.1

−0.6 −0.15 0 0.5 1 −0.1 0 0.1 0.2 x x

(a) Close-up view of O-type grid around the airfoil (b) Magnified view of grid around the leading edge

0.1

0.05

y 0

−0.05

−0.1

−0.15 0.9 1 1.1 x

(c) Magnified view of grid around the trailing edge

Figure 4: O-type grid (elliptic) around an airfoil. The figure illustrates orthogonality and smoothness of the gridlines especially near airfoil surface. where following equation to solve the above Laplace’s equation and obtain 𝜓 at every grid point of the physical domain: 𝛼=𝑥2 +𝑦2, 2 2 𝜂 𝜂 ((𝑥𝜂 +𝑦𝜂)𝜓𝜉𝜉 −2(𝑥𝜉𝑥𝜂 +𝑦𝜉𝑦𝜂)𝜓𝜉𝜂 (17) +(𝑥2 +𝑦2)𝜓 )=0. 𝛽=𝑥𝜉𝑥𝜂 +𝑦𝜉𝑦𝜂, 𝜉 𝜉 𝜂𝜂 (16) 2 2 Tosolvetheaboveequation,thefinitedifferencemethodmay 𝛾=𝑥𝜉 +𝑦𝜉 , be conveniently used.

𝐽=𝑥𝜉𝑦𝜂 −𝑥𝜂𝑦𝜉 (Jacobian of transformation) 4.1. Kutta Condition. The Kutta condition states that the flow leaves the sharp trailing edge of an airfoil smoothly [8]. To apply the Kutta condition in our calculation, we need to 2 2 and ∇ 𝜉=𝑃and ∇ 𝜂=𝑄are control functions which may be consider two possible configurations of the trailing edge. assumedtobezeroinboththegridgenerationandtheflow The trailing edge can have a finite-angle or can be cusped solver sections (𝑃=𝑄=0).Theseassumptionsleadtothe (Figure 5). Journal of Aerodynamics 5

Finite angle From the transformation relationship (see (13)), 1 V 𝜓 = [− (𝑥 )(𝜓 )+(𝑥 )(𝜓 )]. a 2 𝑦 𝐽 𝜂 𝜉 𝜉 𝜂 (21) Airfoil If 𝑉1 and 𝑉𝑁 arethevelocitiesofthegridpoints(1, 1) and V1 (1, 𝑁), respectively, the Kutta condition 𝑉1 =𝑉2 =0gives

At point aV1 =V2 = 0 𝑉1 =𝑉𝑁 =0󳨐⇒𝑢1 =𝑢𝑁 =0, (a) 󵄨 1 󵄨 Cusped [−(𝑥𝜂)(𝜓𝜉)+(𝑥𝜉)(𝜓𝜂)]󵄨 𝐽 󵄨1 Airfoil 󵄨 (22) 1 󵄨 = [−(𝑥𝜂)(𝜓𝜉)+(𝑥𝜉)(𝜓𝜂)]󵄨 =0 𝐽 󵄨𝑁 a 󵄨 −𝑥 𝜓 +𝑥 𝜓 󵄨 =0. 𝜂 𝜉 𝜉 𝜂󵄨1

At point aV1 =V2 ≠ 0 V2 By discretizing (22)inthecomputationaldomain,weget V1 𝑥𝜂𝜓𝜉 =𝑥𝜉𝜓𝜂, (b) [(𝑥 −𝑥 )] [(𝜓 −𝜓 )] Figure 5: Different possible shapes of the trailing edge and their 1,2 1,1 2,1 1,1 relation to the Kutta condition. =[(𝑥2,1 −𝑥1,1)] [(𝜓1,2 −𝜓1,1)] , (23)

y 𝜓 (𝑥 −𝑥 )−𝜓 (𝑥 −𝑥 ) 𝜓 = 2,1 1,2 1,1 1,2 2,1 1,1 . 1,1 𝑥 −𝑥 1, 2 x 1,2 2,1

By considering the wall boundary condition (𝜓1,1 =𝜓1,2), we 1, 1 V 2, 1 Airfoil N can simplify (23)toget 1, N V 2, N 1 𝜓1,1 =𝜓2,1. (24) 1, N − 1 Since the grid points (1, 1) and (1, 𝑁) are the same points in Figure 6: Grid notation of the trailing edge. the physical domain, we have

𝜓1,1 =𝜓1,𝑁 =𝜓2,1. (25)

Suppose that the velocities along the top surface and This value is constant on the airfoil surface due to the wall bottom surface are 𝑉1 and 𝑉2, respectively. For a finite-angle boundary condition. trailing edge, having two finite velocities in two different The derivation of an equation for the cusped trailing edge directions at the same point is physically impossible (Figure is more complicated. Consider the cusped trailing edge and 5(a)) and, therefore, the only possibility is that both velocities the associated grid notation shown in Figure 7. 𝑉 should be zero (𝑉1 =𝑉2 =0). For the cusped trailing edge Since for the cusped trailing edge both vectors 1 and (Figure 5(b)), having two velocities in the same directions at 𝑉𝑁 are equal in the magnitude and direction, from the Kutta 𝑉 =𝑉 point 𝑎 shows that both 𝑉1 and 𝑉2 canbefinite.However,the condition for the cusped trailing edge ( 1 𝑁)wecanwrite 𝑎 pressure at point is unique and Bernoulli equation states 𝑉 =𝑉 󳨐⇒ 𝑢 =𝑢 , that [2] 1 𝑁 1 𝑁 1 1 (26) 1 2 1 2 [ (−𝑥 𝜓 +𝑥 𝜓 )] =[ (−𝑥 𝜓 +𝑥 𝜓 )] . 𝑝 + 𝜌𝑉 =𝑝 + 𝜌𝑉 (18) 𝜂 𝜉 𝜉 𝜂 𝜂 𝜉 𝜉 𝜂 𝑎 2 1 𝑎 2 2 𝐽 1 𝐽 𝑁 or But 󵄨 󵄨 𝑥 󵄨 =𝑥 −𝑥 ,𝑥󵄨 =𝑥 −𝑥 . 𝜉󵄨1 2,1 1,1 𝜉󵄨𝑁 2,𝑁 1,𝑁 (27) 𝑉1 =𝑉2. (19) Since 𝑥2,𝑁 =𝑥2,1 and 𝑥1,𝑁 =𝑥1,1 we have In order to obtain relationships for the Kutta condition in 󵄨 󵄨 terms of stream function 𝜓, consider the finite-angle trailing 𝑥 󵄨 =𝑥󵄨 . 𝜉󵄨1 𝜉󵄨𝑁 (28) edgeintheO-typegridschemeshowninFigure 6. From (1), we have In similar approach, we have 󵄨 󵄨 𝑢=𝜓 . 𝑦 󵄨 =𝑦󵄨 . 𝑦 (20) 𝜉󵄨1 𝜉󵄨𝑁 (29) 6 Journal of Aerodynamics

Cusped NACA 0012, angle of attack = 40

2 1, 2 1.5 2, 1 1, N − 1 1, 1 ( ) 1 1, N (2, N) 0.5 y 0

V1 −0.5 VN −1 Figure 7: Cusped trailing edge and the associated grid notation. −1.5

𝜓 | =𝜓 −𝜓 𝜓 | =𝜓 −𝜓 −2 Furthermore, 𝜉 1 2,1 1,1 and 𝜉 𝑁 2,𝑁 1,𝑁.Since 0 2 𝜓1,1 =𝜓1,𝑁 and 𝜓2,1 =𝜓2,𝑁, x 󵄨 󵄨 𝜓 󵄨 =𝜓󵄨 . 𝜉󵄨1 𝜉󵄨𝑁 (30) Figure 8: Stream function for a finite-angle trailing edge. The figure shows the Kutta condition at the trailing edge. 𝜓 | =𝜓 −𝜓 𝜓 | =𝜓 −𝜓 Moreover, 𝜂 1 1,2 1,1 and 𝜂 𝑁 1,𝑁 1,𝑁−1.Since 𝜓 =𝜓 =𝜓 =𝜓 1,1 1,2 1,𝑁−1 1,𝑁 (wall boundary condition), we NACA 64012, angle of attack = 40 obtain 󵄨 󵄨 2 𝜓 󵄨 =𝜓󵄨 =0. 𝜂󵄨1 𝜂󵄨𝑁 (31)

By substituting (28)through(31)into(26), we have 1 1

(𝑥2,1 −𝑥1,1)(𝑦1,2 −𝑦1,1)−(𝑥1,2 −𝑥1,1)(𝑦2,1 −𝑦1,1) y 0 ×[−(𝑥1,2 −𝑥1,1)(𝜓2,1 −𝜓1,1)+0] 1 = −1 (𝑥2,1 −𝑥1,1)(𝑦1,𝑁 −𝑦1,𝑁−1)−(𝑥1,𝑁 −𝑥1,𝑁−1)(𝑦2,1 −𝑦1,1)

×[−(𝑥1,𝑁 −𝑥1,𝑁−1)(𝜓2,1 −𝜓1,1)+0]. (32) −2

By solving (32)for𝜓1,1(using software Maple), we get −2 0 2 x 𝜓 =𝜓 . 1,1 2,1 (33) Figure 9: Stream function for a cusped trailing edge. The figure shows the Kutta condition at the trailing edge. In addition, 𝜓1,𝑁 =𝜓1,1 =𝜓2,1.Equation(33)istherequired expression for the cusped trailing angle. Figures 8 and 9 show the stream function 𝜓 for both the ∘ The velocity values on the outer boundaries are known from finite-angle (NACA 0012 airfoil with angle of attack of 𝛼=40 the conditions at infinity (using (1)and(2)). In other words, and a free stream velocity of 𝑉∞ =70m/s) and the cusped ∘ 𝑥-component of the velocity vector (𝑢)onalltheouter (NACA 64012 with angle of attack of 𝛼=40 and a free stream boundaries is equal to 𝑉∞ cos 𝛼 and 𝑦-component of the velocity of 𝑉∞ =70m/s) trailing edge, respectively. velocity vector (V) on all the outer boundaries is equal to 𝑉∞ sin 𝛼. For the inside of the physical domain and the airfoil 4.2. Velocity Calculation. There are three sections where the surface, we can use (12)and(13) as follows: velocity must be known: 󵄨 𝜕𝜓󵄨 1 𝐶𝐷 𝐷𝐸 𝐸𝐹 𝐹𝐶 𝑢𝑖,𝑗 = 󵄨 = [−(𝑥𝜂) (𝜓𝜉) +(𝑥𝜉) (𝜓𝜂) ], (1) the outer boundaries (four sides , , ,and 𝜕𝑦 󵄨 𝐽 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 of the rectangle shown in Figure 2), 𝑖,𝑗 󵄨 (34) 𝐴𝐻 𝜕𝜓󵄨 1 (2) the airfoil surface ( in Figure 2), V =− 󵄨 =− [(𝑦 ) (𝜓 ) −(𝑦) (𝜓 ) ]. 𝑖,𝑗 𝜕𝑥 󵄨 𝐽 𝜂 𝑖,𝑗 𝜉 𝑖,𝑗 𝜉 𝑖,𝑗 𝜂 𝑖,𝑗 (3) the inside of the physical domain. 󵄨𝑖,𝑗 Journal of Aerodynamics 7

The central and one-sided difference schemes are usedfor −6 the inside of the physical domain and the airfoil surface, respectively. After obtaining the components of the velocity −5 vector,thetotalvelocitycanbecomputedby −4 2 2 𝑉𝑖,𝑗 = √𝑢𝑖,𝑗 + V𝑖,𝑗. (35) −3

As stated before, for an incompressible flow, the pressure p C coefficient can be expressed in terms of velocity only. Thus −2 (4) can be used to determine the pressure of any grid point in the domain. Therefore, −1 1 𝑝 = 𝜌(𝑉2 −𝑉2 )+𝑝 . 𝑖,𝑗 2 ∞ 𝑖,𝑗 ∞ (36) 0

1 4.3.KuttaConditioninTermsoftheVelocityPotential. The proposed method can be easily developed in terms of the 0 0.2 0.4 0.6 0.8 1 velocity potential 𝜙.Thewallboundaryconditionmaybe x/c expressed in terms of either the velocity potential 𝜙, (𝜕𝜙/𝜕𝑛 = 0),orthestreamfunction𝜓, (𝜕𝜓/𝜕𝑠,where =0) 𝑛 and Results from reference 𝑠 aretheunitvectornormaltotheairfoilsurfaceandthe Results from our method distance along the body (airfoil) surface, respectively. Using Airfoil: NACA 0012 the transformation relationships for mapping the physical Angle of attack: 9 domain onto the computational one, we can write Figure 10: Comparison between the results from [2]andtheresults 𝜕𝜙 −1 from our method for validation case 1. The figure shows an excellent = (𝛼𝜙 −𝛽𝜙 )=0, 𝜕𝑛 𝐽√𝛼 𝜉 𝜂 (37) agreement between the results. airfoil surface where 𝐽, 𝛼,and𝛽 aredefinedin(16). The solution of the above equation for the airfoil surface using the finite difference 5.2. Trailing Edge with Finite-Angle method gives the value for 𝜙1,𝑗 (𝑗 = 1,...,𝑁). From the Validation Case 1. The pressure coefficient distribution (𝐶𝑝) definition of the velocity potential, ∘ overaNACA0012airfoilatanangleofattackof𝛼=9 is V =𝜙𝑦. (38) plotted. The results are compared with the results from [2]. The O-type grid size used in the computation is 155 × 155. In a similar way to the derivation for Kutta condition in terms The computation time is 53 seconds (see Figure 10). of the stream function given in (21)to(23), we get Validation Case 2. The pressure coefficient distribution (𝐶𝑝) 1 ∘ 𝜙 = [− (𝑥 )(𝜙 )+(𝑥 )(𝜙 )], over a NACA 0024 airfoil at an angle of attack of 𝛼=0 is 𝑦 𝐽 𝜂 𝜉 𝜉 𝜂 (39) plotted. The results are compared with the results from [9]. 155 × 155 𝑉1 =𝑉𝑁 =0󳨐⇒V1 = V𝑁 =0. The O-type grid size used in the computation is . The computation time is 41 seconds (see Figure 11). And finally Validation Case 3. The pressure coefficient distribution (𝐶𝑝) 𝜙 (𝑥 −𝑥 )−𝜙 (𝑥 −𝑥 ) ∘ 2,1 1,2 1,1 1,2 2,1 1,1 overaNACA4414airfoilatanangleofattackof𝛼=2 𝜙1,1 = . (40) 𝑥1,2 −𝑥2,1 is plotted. The results are compared with the results from the software Xfoil [10]. The O-type grid size used in the By including (37)and(40)intothesolutionloops,wecanfind computation is 155×155. The computation time is 51 seconds the velocity potential over the domain. The above procedure (see Figure 12). also can be extended to the three dimension case. Validation Case 4. The pressure coefficient distribution (𝐶𝑝) ∘ 5. Results over a NACA 4412 airfoil at an angle of attack of 𝛼=10 is plotted. The results are compared with the results in[5]. The 5.1. Validation of the Results for the Pressure Distribution. The O-type grid size used in the computation is 155 × 155.The resultsobtainedherearecomparedwiththeresultsfrom computation time is 55 seconds (see Figure 13). usingthepanelmethod.TheresultsareobtainedbyaFortran compiler (PGI) and computations are run on a PC with Intel 5.3. Cusped Trailing Edge Pentium Dual 1.73 and 1 G RAM. The tolerance used in the iterative loops (the mesh generation and the stream function) Validation Case 1. The pressure coefficient distribution (𝐶𝑝) −8 ∘ is 10 . over a NACA 64012 airfoil at an angle of attack of 𝛼=6 8 Journal of Aerodynamics

1

0.5 p C 0

−0.5

−1 0 0.2 0.4 0.6 0.8 1 x/c

Results from reference Results from our method Airfoil: NACA 0024 Angle of attack: 0

Figure 11: Comparison between the results from [9] and the results from our method for validation case 2. The figure shows an excellent agreement between the results.

−1

−0.5 p C 0

0.5

1

0 0.2 0.4 0.6 0.8 1 x/c

Results from XFoil Results from our method Airfoil: NACA 4414 Angle of attack: 2

Figure 12: Comparison between the results from [10] and the results from our method for validation case 3. The figure shows an excellent agreement between the results.

is plotted. The results are compared with the results from Excellent agreement can be obtained by comparing the the software XFLR5 [11]. The O-type grid size used in the results from our method and the ones from the panel method computation is 245 × 245. The computation time is 4 minutes giveninvalidationcasesforbothfinite-angleandthecusped and 15 seconds (see Figures 14, 15,and16). trailing edges. As shown in the validation cases results, the Journal of Aerodynamics 9

−6 −3.5 −3.0 −2.5 −2.0 −5 p

C −1.5 −1.0 −0.5 −4 0 0.5 0 0.2 0.4 0.6 0.8 1.0 −3 x p C −2 NACA 64012-Re = 100000-alpha = 6.00 inviscid

Fixed speed polar −1 Reynolds = 100 000 Mach = 0.000 NCrit = 9.000 Forced upper trans. = 1.000 Forced lower trans. = 1.000 0 ∘ NACA 64012 Alpha = 6.00 12.00 Thickness = % C1 = 0.619 37.40 Max. thick. pos. = % Cm = 0.009 Max. camber = −0.00% 1 Cd = 0.000 Max. camber pos. = 44.20% 0.000 Number of panels = 245 Upper trans. = Lower trans. = 0.000 0 0.2 0.4 0.6 0.8 1 Figure 15: Pressure coefficient distribution over a NACA 64012 x/c ∘ airfoil at a 6 angle of attack obtained by XFLR5. Results from reference Results from our method −4 Airfoil: NACA 4412 Angle of attack: 10

Figure 13: Comparison between the results from [5]andtheresults −3 from our method for validation case 4. The figure shows an excellent agreement between the results. −2 p C

103000 −1 102000 1 101000 100000 0 0.5 99000 y 98000 1

97000 (Pa) P 0 96000 0 0.2 0.4 0.6 0.8 1 95000 x/c 94000 −0.5 Results from XFLR5 V6.10 93000 Results from our method 92000 Airfoil: NACA 64012 (using 245 points) 6 −0.5 0 0.5 1 1.5 Angle of attack: x Figure 16: Comparison between the results from the software Figure 14: Pressure distribution over the airfoil (NACA 64012) used XFLR5andtheresultsfromourmethodforvalidationcase1(cusped in validation case 1 (cusped trailing edge). trailing edge). The figure shows an excellent agreement between the results.

maximum value for 𝐶𝑝 is exactly equal to 1. The pressure 6. Conclusion coefficientatthetrailingedge(T.E.)isequaltounitybecause thevelocityiszeroatthisstagnationpoint.Accordingly, This paper presents a novel method to implement the Kutta condition in the numerical solution of two-dimensional 2 incompressible potential flow over an airfoil. The proposed 𝑉 . . 𝑉 =0, 𝐶 =1− T E =1−0=1. (41) methodisbasedonsolvingtheLaplace’sequationforthe T.E. 𝑝 𝑉2 ∞ stream function at each grid point generated by the elliptic grid generation technique (O-type). Therefore, it is exempt 𝑉 =0̸ 𝐶 For the cusped trailing edge, T.E. .Thusthevalueof 𝑝 at from considering the panels and the quantities such as the T.E. is not equal to 1 (𝐶𝑝 =1̸ ), as shown in Figure 16. vortex panel strength and circulation used in the panel 10 Journal of Aerodynamics

method. It applies for both finite-angle and cusped trailing edges. A novel and very easy to implement expression for the stream function for the finite-angle and the cusped trailing edges is derived. The accurate results obtained for both cases show the correctness and accuracy of the numerical scheme.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] J. L. Hess and A. M. O. Smith, “Calculation of potential flow about arbitrary bodies,” Progress in Aerospace Sciences,vol.8, pp. 1–138, 1967. [2]J.D.Anderson,Fundamentals of Aerodynamics, McGraw-Hill, New York, NY, USA, 2001. [3] J. Katz and A. Plotkin, Low-Speed Aerodynamics, Cambridge University Press, New York, NY, USA, 2001. [4] J. L. Hess, “Panel methods in computational fluid dynamics,” Annual Review of Fluid Mechanics,vol.22,no.1,pp.255–274, 1990. [5] R. L. Fearn, “Airfoil aerodynamics using panel methods,” The Mathematica Journal,vol.10,no.4,2008. [6] J. Hess, “Development and Application of Panel Methods,” in Advanced Boundary Element Methods,T.Cruse,Ed.,pp.165– 177, Springer, Berlin, Germany, 1988. [7]J.F.Thompson,F.C.Thames,andC.W.Mastin,“Automatic numerical generation of body-fitted curvilinear coordinate sys- tem for field containing any number of arbitrary two-dimen- sional bodies,” Journal of Computational Physics,vol.15,no.3, pp.299–319,1974. [8]M.W.Kutta,Lifting Forces in Flowing Fluids,1902. [9] E. L. Houghton and P. W. Carpenter, Aerodynamics For Engi- neering Students, Elsevier Science, Amsterdam, The Nether- lands, 2012. [10] M. Drela, “XFOIL: an analysis and design system for low reyn- olds number airfoils,” in Low Reynolds Number Aerodynamics, vol. 54 of Lecture Notes in Engineering, pp. 1–12, Springer, Berlin, Germany, 1989. [11] XFLR5: an analysis tool for airfoils, and planes, http:// www.xflr5.com/xflr5.htm. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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