================================================================================ Airfoil generator utilizing the Joukowski transformation Written by: Russell Leighton 762 1/2 W. Newgrove Lancaster, CA 93534 22 March 1987 Addendum by: David Foster 1060 Hemlock Drive Rochester, MI 48063 19 June 1988 ================================================================================ The following paper is a condensed version of the paper I originally wrote describing the Joukowski tranformation. It was submitted to the Aeronautical and Mechanical Engineering Department, School of Engineering and Technology of the California Polytechnic State University in San Luis Obispo, CA June 1984. ================================================================================ A Computational Flow Visualization Technique Utilizing the Joukowski Transformation Written by: Russell A. Leighton LIST OF SYMBOLS: u - Real component in w-plane v - Imaginary component in w-plane w - Complex resultant plane x - Real component in z-plane y - Imaginary component in z-plane z - Complex source plane i - Imaginary unit (square root of -1) *** U - Free stream velocity r - Circle radius alpha - angle of incidence *** INTRODUCTION: Conformal mapping is a very useful mathematical tool and has applications in the engineering field. One particular application utilizes conformal mapping to simplify the visualization of fluid flow about airfoil sections. By simplfing the mathematics this technique allows for faster computation and therefore could be used for real time computational flow visualization. The potential uses for a computational flow visualization technique range from an educational tool, illustrating the behavior of fluid flow about airfoils, to an advanced modeling tool. Since the actual flow is computed about a simple shape, the circle, development of the flow visualization equations is straight forward. The following sections will discuss the development of the mapping equations, the streamline and pressure distribution equations, and reverse mapping equations necessary for the calculation of the circle parameters given airfoil data. DISCUSSION: Conformal Mapping ----------------- Conformal mapping is a mathematical tool that can be used to visualize the nature of complex functions. A definition of conformal mapping can be understood by picturing two distinct planes, the source plane (z-plane) and the resultant plane (w-plane). Given a domain D of the z-plane and a complex function, w = f(z) relating the z-plane to the w-plane, for each point in domain D there exists a corresponding point in the w-plane. If the function, f(z) is an analytic function then the mapping given by f(z) is said to be conformal, or angle-preserving, except at points where the derivative, f'(z) is zero. The general form of the complex function relating the z-plane to the w-plane is: (1) w = f(z) = u(x,y) + (i)v(x,y) where: z = x + (i)y The point wo = f(zo) corresponding to a point zo is called the "image" of the point zo with respect to the mapping defined by f(z). A set of points representing a function in the z-plane will have a corresponding set of points, or "image" in the w-plane. Points located in the z-plane, such that the derivative of the mapping function goes to zero, are called critical points. At these points the mapping is said to be non-conformal (i.e. the angles are not preserved). As will be shown, these points are important for the following mapping. The Joukowski Transformation ---------------------------- The following mapping function is important in the field of aerodynamics because of the nature of its' transformation. With this mapping function if a circle is plotted in the z-plane, such that its' center is near the origin and it passes through one critical point, it will be transformed into an airfoil shape. The form of this function is: (2) w = z + 1/z Its' derivative is: (3) w' = 1 - 1/z = (z + 1)(z - 1)/z Therefore, the mapping will be conformal except at points z = 1 and z = -1, where w' goes to zero. If plotted it would be evident that passing through one of these points will produce a sharp edge resembling the trailing edge of an airfoil. If the geometry of a circle is such that one of the critical points is intersected while the other is bypassed, an airfoil shape will result from the transformation. This transformation is commonly known as the Joukowski transformation which was named for the Russian mathematician, Nikolai Jegorovich Joukowski for his initial use of this mapping function. *** See below for second critical point location *** Computer Implementation ----------------------- The derivation of the equations suitable for computer implementation is as follows. Given the complex function: (4) w = u + (i)v = z + 1/z where u is the horizontal component in the w-plane and v is the vertical component. If (5) z = x + (i)y then (6) 1/z = [1/(x + (i)y)][(x - (i)y)/(x - (i)y)] Separate the real and imaginary parts to obtain (7) 1/z = (x/s) - (i)(y/s) where: s = x^2 + y^2 therefore, from equation (4) (8) u = x + x/s (9) v = y - y/s where: s = x^2 + y^2 These equations define the mapping process and can be easily implemented into computer software (see C source listing). To define the circle in the z-plane the radius and the location of its' center are necessary. Since the circle must pass through one of the two critical points and bypass the other it is necessary that the radius be greater than one. This is actually more information than is required to define the circle. For example, one component of the circle center location could be calculated from the other component, the radius, and the known critical point (e.g. -1,0). Likewise, any of the other parameters may be calculated if the remaining parameters are known. The Inverse Mapping ------------------- It has been shown that airfoil shapes may be easily obtained from the Joukowski transformation of the relativily simple shape, the circle. However, it is not convenient to define these airfoil shapes in terms of their corresponding circle parameters (the radius and center location). To determine the necessary circle parameters, an inverse mapping (or a mapping from the w-plane to the z-plane) may be performed. Two airfoil parameters, the camber and thickness, are useful for defining the airfoil. A very simple inverse mapping, requiring only three points to be mapped, can be found by specifing the camber and thickness at the mid-chord location. The derivation of this inverse mapping is rather involved, therefore, it is left to the reader to determine, if so interested (or just take a look at the C source listing and try to figure it out). Flow About Cylinders and Airfoils --------------------------------- The usefulness of the Joukowski transformation is derived mostly from the fact that a circle is a much simpler shape than the airfoil section. This property of this particular mapping can be further exploited by recognizing that not only is the airfoil exactly represented by the circle (or a unit depth cylinder), but the region about the airfoil is also represented by the region surrounding the cylinder. This means that any curves plotted about the cylinder, in the z-plane, have corresponding curves located about the airfoil, in the w-plane. Specifically, streamline and pressure distribution plots may be computed for the cylinder and then mapped onto the w-plane in order to obtain the corresponding streamline and pressure distribution plots about the airfoil. Again the equations for the streamline and pressure distribution plots can be derived by the reader if so interested (the theory can be found in most aeronautical engineering references). *** * Also Advanced Engineering Mathematics,C.R.Wylie, pp 416-428 * McGraw Hill *** Angle of Attack and Rotation Tranformation ------------------------------------------ The angle of attack may be included in the equations describing the flow about the cylinder. It is interesting to note that any changes in angle of attack will not result in any change in the flow about the cylinder except that the angle at which the flow enters the region about the cylinder should be equal to the negative value of the angle of attack. A simple rotation transformation would bring the flow direction back to the horizontal, resulting in no apparent change from a zero angle of attack. It should be noted, however, that the local coordinate axis is no longer coincident with the global coordinate axis. Because of this difference the Joukowski transformation will result in an airfoil at an angle of attack with the flow direction coming into and leaving the region of the airfoil, parallel to the horizontal global coordinate axis. *** * Addendum * It will be apparent, looking at the original version of the program that * the streamlines obtained are not realistic at the trailing edge when * the airfoil is at other than zero incidence. Also, the pressure plot * reveals that no lift is generated, because the pressure is equal * above and below the airfoil. * This deficiency has long been recognized, and the standard correction * is to add the complex potential for a point vortex to the original * flow. * The added term is * -K.i.log(z)/2.PI * which results in an addition to the stream function of * K.log(rs/r)/2.PI * but does not change the value of zero for the circle and 'dividing' * streamline, since at rs = r, log(rs/r) = log (1) = 0 * The Joukowski hypothesis is that the circulation K is such that the * second stagnation point is at the point on the circle which will map * into the trailing edge of the airfoil.In terms of the incidence, it * results that * K = 4.PI.U.r.sin(alpha) *** The same is also true for the pressure distribution. At any given angle of attack, the pressure distribution will remain the same for the cylinder. *** * When the circulation is added, the pressure distribution for both the * circle and the airfoil are now non-symmetric. This is a central * feature of the transformation, and can be shown to result in the * KUTTA - JOUKOWSKI LAW * Lift = Density.U.K. *** However, once transformed to the w-plane, the resulting pressure distribution will be about an airfoil at the given angle of attack. *** * In the code airfoil.c * * the log(rs/r) term has been approximated by its first order expansion * derived from log(R) = 2.{ (R-1)/(R+1) + ... <[(R-1)/(R+1)]^n>/n ... } * This has enabled the elegant plotting scheme devised by the original * author to be retained, while including the essential features of the * circulation in correcting the streamlines and pressure distribution *** Possible Additions to the Model ------------------------------- The equations for the streamline plot and the pressure distribution are easily derived for flow about a simple cylinder. The equations, or model used in the program assume invisid, irrotational flow and were therefore the simplest to derive. A possible addition to this model would be to incorporate boundary layer effects into the equations describing the flow about the cylinder. Another addition, that is important if precise airfoil geometry is required, is the incorporation of a complete inverse mapping capability. A complete inverse mapping would allow for a point by point description of the airfoil as input to the model. This airfoil geometry would, in turn, be mapped from the w-plane onto the z-plane resulting in an approximate cylindrical shape. The flow model may then be developed for this approximate cylinder and the corresponding flow model, describing the flow about the airfoil, may then be obtained by the forward mapping process. Although complex this addition would increase the accuracy of this modeling technique giving results suitable for comparison to experimental results. CONCLUSION: By simplifing the modeling process, conformal mapping and in particular, the Joukowski transformation, offers a simple and fast method for computational flow visualization of fluid flow about arbitrary airfoil sections. The equations necessary for the mapping process are readily incorporated into a computer program which aids in the production of a graphical output of the transformation. The potential use of the Joukowski transformation is only limited by the fluid model developed to describe the flow about the cylinder. Since the modeling process is simplified, complex fluid models can be more easily incorporated.