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On the Kutta Condition in Potential Flow Over Airfoil Hindawi Publishing Corporation Journal of Aerodynamics Volume 2014, Article ID 676912, 10 pages http://dx.doi.org/10.1155/2014/676912 Research Article On the Kutta Condition in Potential Flow over Airfoil 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 airfoils 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 fluid dynamics 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 vortex 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 circulation. 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 is defined as 2 −∞ = constant V∞ ≡ =1−( ) . V∞ 2 (4) (1/2) ∞∞ ∞ At standard sea level conditions, 3 5 2 ∞ = 1.23 kg/m ,∞ = 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 and physical plane (Figure 2)ontothe and (Figure 1). The flow is governed by the Laplace’s equation computational plane (Figure 3), which is a regular region. It 2 ∇ =0( isthestreamfunction).Theboundaryconditions is based on solving the Poisson equations as follows: are as shown in Figure 1. + =(,), (6) 2.1. Conditions at Infinity. Far away from the airfoil surface + =(,), (toward infinity), in all directions, the flow approaches the uniform free stream conditions. If the angle of attack (AOA) where and are the computational coordinates correspond- is and the free stream velocity is ∞, then the components ing to and in the physical coordinate, respectively. and oftheflowvelocitycanbewrittenas are grid control functions which control the density of grids towards a specified coordinate line or about a specific grid = = , ∞ cos (1) point. To find an explicit relation for and in terms of grid points (∈[1,])and (∈[1,]), the following relations may be used: V =− = , (2) ∞ sin −2 + =−2 ( (, ) +(,) ), V V where and are components of velocity vector ;thatis, (7) V =i + Vj i j 2 ( and are the unit vectors in and directions, −2 + =− ( (, ) +(,)), resp.). where 2.2. Condition on the Airfoil Surface. For inviscid flow, flow =2 +2, cannot penetrate the airfoil surface. Thus the velocity vector must be tangent to the surface. This wall boundary condition = + , canbeexpressedby (8) 2 2 = + =0 = , (3) or constant = − (Jacobian of transformation) . where is tangent to the surface. In the problem of the flow over an airfoil, if the free stream velocity ∞ and the angle of The solution of the above equations (using the finite dif- attack are known, from the boundary conditions at infinity ference method to discretize the terms) gives and (see (1)and(2)) and the wall boundary condition (see (3)) coordinates (in the physical domain) of coordinate (, ) in one can compute the stream function at any point of the the computational domain. physical domain (flow region). Then, by knowing ,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 ∞ and are known, the stream function at any point N=2N1 +2N2 +N3 +6 in the physical domain can be obtained from (1)and(2)as N1 follows: = +( −)∞ cos , A (1, 1) M B (M, 1) (9) N3 H(1, N) M G (M, N) = −( −)∞ sin , where subscripts and 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 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 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 with respect to the position variables and .By usingthechainrule,itcanbeconcludedthat = + = + , (1, N) G (M, N) H M (10) F (M,N1 +2N2 +N3 +5) = + = + . E (M,N1 +N2 +N3 +4) N = 2N1 +2N2 +N3 +6 N D (M,N1 +N2 +3) By interchanging and ,and and ,thefollowing relationships can also be derived: C (M,N1 +2) = + = + , (1, 1) A M B (M, 1) (11) = + = + . Figure 3: The computational domain showing the discretization of the physical domain boundaries. By solving (11)for/ and /,wefinallyobtain 1 = ( − ) , (12) 1 advantage that the grid around the airfoil is orthogonal. The = (− + ) , discretization of the physical domain and the corresponding (13) computational domain are shown in Figures 2 and 3,respec- =2 +2 + tively. In the computational domain, and 1 2 where = − is Jacobian of the
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