Kármán–Trefftz Airfoil
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In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky (who published it in 1910), is a
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
historically used to understand some principles of
airfoil An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
design. The transform is : z = \zeta + \frac, where z = x + iy is a complex variable in the new space and \zeta = \chi + i \eta is a complex variable in the original space. This transform is also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations. In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
(z-plane) by applying the Joukowsky transform to a circle in the \zeta-plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the point \zeta = -1 (where the derivative is zero) and intersects the point \zeta = 1. This can be achieved for any allowable centre position \mu_x + i\mu_y by varying the radius of the circle. Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the Kármán–Trefftz transform, generates the broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.


General Joukowsky transform

The Joukowsky transform of any complex number \zeta to z is as follows: :\begin z &= x + iy = \zeta + \frac \\ &= \chi + i \eta + \frac \\ pt &= \chi + i \eta + \frac \\ pt &= \chi\left(1 + \frac1\right) + i\eta\left(1 - \frac1\right). \end So the real (x) and imaginary (y) components are: :\begin x &= \chi\left(1 + \frac1\right), \\ pt y &= \eta\left(1 - \frac1\right). \end


Sample Joukowsky airfoil

The transformation of all complex numbers on the unit circle is a special case. , \zeta, = \sqrt = 1, which gives \chi^2 + \eta^2 = 1. So the real component becomes x = \chi (1 + 1) = 2\chi and the imaginary component becomes y = \eta (1 - 1) = 0. Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2. Transformations from other circles make a wide range of airfoil shapes.


Velocity field and circulation for the Joukowsky airfoil

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of
uniform flow In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
, a doublet, and a vortex. The complex conjugate velocity \widetilde = \widetilde_x - i\widetilde_y, around the circle in the \zeta-plane is \widetilde = V_\infty e^ + \frac - \frac, where * \mu = \mu_x + i \mu_y is the complex coordinate of the centre of the circle, * V_\infty is the freestream velocity of the fluid, \alpha is the
angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
of the airfoil with respect to the freestream flow, * R is the radius of the circle, calculated using R = \sqrt, * \Gamma is the
circulation Circulation may refer to: Science and technology * Atmospheric circulation, the large-scale movement of air * Circulation (physics), the path integral of the fluid velocity around a closed curve in a fluid flow field * Circulatory system, a bio ...
, found using the Kutta condition, which reduces in this case to \Gamma = 4\pi V_\infty R\sin\left(\alpha + \sin^\frac\right). The complex velocity W around the airfoil in the z-plane is, according to the rules of conformal mapping and using the Joukowsky transformation, W = \frac = \frac. Here W = u_x - i u_y, with u_x and u_y the velocity components in the x and y directions respectively (z = x + iy, with x and y real-valued). From this velocity, other properties of interest of the flow, such as the
coefficient of pressure The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow field has its own ...
and lift per unit of span can be calculated. A Joukowsky airfoil has a cusp at the trailing edge. The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically been romanized in a number of ways, thus the variation in spelling of the transform.


Kármán–Trefftz transform

The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil—which is the result of the transform of a circle in the \zeta-plane to the physical z-plane, analogue to the definition of the Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle \alpha. This transform is where b is a real constant that determines the positions where dz/d\zeta = 0, and n is slightly smaller than 2. The angle \alpha between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to n as : \alpha = 2\pi - n\pi, \quad n = 2 - \frac. The derivative dz/d\zeta, required to compute the velocity field, is : \frac = \frac \frac .


Background

First, add and subtract 2 from the Joukowsky transform, as given above: : \begin z + 2 &= \zeta + 2 + \frac = \frac (\zeta + 1)^2, \\ pt z - 2 &= \zeta - 2 + \frac = \frac (\zeta - 1)^2. \end Dividing the left and right hand sides gives : \frac = \left( \frac \right)^2. The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near \zeta = +1. From conformal mapping theory, this quadratic map is known to change a half plane in the \zeta-space into potential flow around a semi-infinite straight line. Further, values of the power less than 2 will result in flow around a finite angle. So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp. Replacing 2 by n in the previous equation gives : \frac = \left( \frac \right)^n, which is the Kármán–Trefftz transform. Solving for z gives it in the form of equation .


Symmetrical Joukowsky airfoils

In 1943 Hsue-shen Tsien published a transform of a circle of radius a into a symmetrical airfoil that depends on parameter \epsilon and angle of inclination \alpha: : z = e^ \left(\zeta - \epsilon + \frac + \frac\right). The parameter \epsilon yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil. Furthermore the radius of the cylinder a=1+\epsilon.


Notes


References

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External links


Joukowski Transform NASA Applet

Joukowsky Transform Interactive WebApp
Conformal mappings Aerodynamics Aircraft wing design