Lecture 19 Notes: Stream Functions and Conformal Maps

18.354J Nonlinear Dynamics II: Continuum Systems 19 Lecture 19 Stream functions and conformal maps There is a useful device for thinking about two dimensional flows, called the stream function of the flow. The stream function ψ(x, y) is defined as follows   ∂ψ ∂ψ u = (u, v) = ,− . (469) ∂y ∂x The velocity field described by ψ automatically satisfies the incompressibility condition, and it should be noted that u · ∇ψ = u ∂ψ ∂ψ +v = 0. ∂y ∂x (470) Thus ψ is constant along streamlines of the flow. Besides it’s physical convenience, another great thing about the stream function is the following. By definition ∂ψ ∂φ = , ∂y ∂x ∂ψ ∂φ v = − = , ∂x ∂y u = (471a) (471b) where φ is the velocity potential for an irrotational flow. Thus, both φ and ψ obey the well known Cauchy-Riemann equations of complex analysis. 19.1 The Cauchy-Riemann equations In complex analysis you work with the complex variable z = x + iy. Thus, if you have some complex function f (z) what is df /dz? Well, f (z) can be separated into a real part u(x, y) and an imaginary part v(x, y), where u and v are real functions, i.e.: f (z) = f (x + iy) = u(x, y) + iv(x, y). (472) For example, if f (z) = z 2 then u = x2 − y 2 and v = 2xy. What then is df /dz? Since we are now in two-dimensions we can approach a particular point z from the x-direction or the y-direction (or any other direction, for that matter). On one hand we could define df ∂f ∂u ∂v = = +i . ∂x dz ∂x ∂x (473a) df ∂f ∂f ∂u ∂v = = −i = −i + . dz ∂(iy) ∂y ∂y ∂y (473b) Or, alternatively For the definition of the derivative to make sense requires ∂u/∂x = ∂v/∂y and −∂u/∂y = ∂v/∂x, the Cauchy-Riemann equations. If this is true then f (z) is said to be analytic and we can simply differentiate with respect to z in the usual manner. For our simple example f (z) = z 2 we have that df /dz = 2z (confirm for yourself that z 2 is analytic as there are many functions that are not, e.g., |z| is not an analytic function.) 19.2 Conformal mapping We can now use the power of complex analysis to think about two dimensional potential flow problems. Since φ and ψ obey the Cauchy-Riemann equations, this implies that w = φ + iψ is an analytic function of the complex variable z = x + iy. We call w the complex potential. Another important property of 2D incompressible flow is that both φ and ψ satisfy Laplace’s equation. For example, using the Cauchy-Riemann equations we see that ∂ψ ∂ψ ∂2ψ ∂2ψ + = − + = 0. ∂x2 ∂y 2 ∂x∂y ∂y∂x (474) The same proof can be used for φ. We can therefore consider any analytic function (e.g., sin z,z 4 ,...), calculate the real and imaginary parts and both of them satisfy Laplace’s equation. The velocity components u and v are directly related to dw/dz, which is conveniently calculated as follows: dw ∂φ ∂ψ = u − iv. (475) = +i ∂x dz ∂x As a simple example consider uniform flow at an angle α to the x-axis. The corresponding complex potential is w = u0 ze−iα . In this case dw/dz = u0 e−iα . Using the above relation, this tells us that u = u0 cos α and v = u0 sin α. We can also determine the complex potential for flow past a cylinder since we know that   R2 φ = u0 r + cos θ, (476) r and this is just the real part of the complex potential   R2 . w = u0 z + z (477) Check this by substituting in z = reiθ . What is the corresponding stream function? Also w(z) = −i ln z is the complex potential for a point vortex since   Re (w(z)) = Re −i ln(reiθ ) = θ, (478) and we know that φ = θ is the real potential for a point vortex. Thus   R2 iΓ w(z) = u0 z + − ln z z 2π (479) is the complex potential for flow past a cylinder with circulation Γ. So let’s assume that the only problem we know how to solve is flow past a cylinder, when really we want to know how to solve for flow past an aerofoil. The idea is to now consider two complex planes (x, y) and (X, Y ). In the first plane we have the complex variable z = x + iy and in the latter we have Z = X + iY . If we construct a mapping Z = F (z) which is analytic, with an inverse z = F −1 (Z), then W (Z) = w(F −1 (Z)) is also analytic, and may be considered a complex potential in the new co-ordinate system. Because W (Z) and w(z) take the same value at corresponding points of the two planes it follows that Ψ and ψ are the same at corresponding points. Thus streamlines are mapped into streamlines. In particular a solid boundary in the z-plane, which is necessarily a streamline, gets mapped into a streamline in the Z-plane, which could accordingly be viewed as a rigid boundary. Thus all we have done is distort the streamlines and the boundary leaving us with the key question: Given flow past a circular cylinder in the z-plane can we choose a mapping so as to obtain in the Z-plane uniform flow past a more wing-like shape? (Note that we have brushed passed some technical details here, such as the requirement that dF/dz 6= 0 at any point, as this would cause a blow-up of the velocity). 19.3 Simple conformal maps The simplest map is Z = F (z) = z + b, (480) which corresponds to a translation. Then there is Z = F (z) = zeiα , (481) which corresponds to a rotation through angle α. In this case, the complex potential for uniform flow past a cylinder making angle α with the stream is   R2 iα iΓ −iα W (Z) = u0 Ze + e − ln Z. (482) Z 2π Note, that this expression could also include the term ln eiα = iα which I have neglected. This is just a constant however and doesn’t change the velocity. Finally there is the non-trivial Joukowski transformation, Z = F (z) = z + c2 . z (483) What does this do to the circle? Well, z = aeiθ becomes Z = aeiθ + c2 c2 c2 −iθ e = (a + ) cos θ + i(a − ) sin θ. a a a (484) Defining X = Re(Z), Y = Im(Z), it is easily shown that !2 X a+ c2 a + !2 Y a− c2 a which is the equation of an ellipse, provided c < a. = 1, (485)