# Generation of lift and uniqueness for 2D Laplace equation around body

I am studying Anderson Fundamentals of Aerodynamics on my own (as a physics major), and have been struggling to understand the distinction between lifting and non-lifting flow (in particular, around a cylinder) and some of the heuristics he uses.

Anderson begins by solving Laplace equation around a cylinder (i.e. $D=\mathbb{R}^2/B_1$) subject to Neumann conditions at the outer and cylinder boundary ($d\phi/dn$=0 at cylinder, $d\phi/dx=V_\infty$, $d\phi/dy=0$) with a combined dipole/doublet and linear potential

$$\phi_1=V_\infty r \cos\theta +\frac{\kappa}{2\pi}\frac{\cos\theta}{r}$$

but then constructs a "lifting" potential by adding a vortex flow,

$$\phi_2=V_\infty r \cos\theta +\frac{\kappa}{2\pi}\frac{\cos\theta}{r}-\frac{\Gamma}{2\pi}\theta$$

which also satisfies the prescribed boundary conditions. He explains that one flow is produced by a rotating cylinder, and the other by a stationary cylinder, but I am struggling to understand

1. How two potentials can exist which satisfy the prescribed boundaries (indeed an infinite number, since ΓΓ is arbitrary) given the uniqueness theorem for Neumann boundaries? Does the uniqueness theorem only hold for connected domains? This is a silly question, but one I haven't been able to answer.

2. How the rotating cylinder produces the vortex flow at all? Anderson has been neglecting viscous effects entirely in this chapter, but here seems to imply that the lift generated by the rotating cylinder is due to friction/viscous forces. The flow is clearly irrotational for $r\neq 0$, but he has emphasized elsewhere that viscous forces produce rotational flow, so I'm struggling to reconcile these points.

3. Later in the book, he solves the problem by constructing potential exclusively using "vortex sheets" to satisfy the Neumann boundary, so I'm wondering what the significance of the totally irrotational flows (the dipole and inverse flows) is in the context of circulation/flow theory? If we can satisfy the boundary conditions without vortex flow, what theoretic constraint makes us use them? It's likely related to the Kutta condition, but in an inobvious way.

Edit: follow-up questions:

1. When you say the boundary conditions aren't given everywhere, do you mean that the "implicit" Neumann boundaries (that $\mathbf{V}$ must go to $\mathbf{V}_\infty$ as $x,y\rightarrow\infty$) are not sufficient to guarantee uniqueness? Looking at a proof of the uniqueness theorem for the Laplace equation using the maximum principle, this kind of implicit boundary doesn't really seem to satisfy the uniqueness conditions at all. I'm wondering what effect those conditions have on constraining the set of possible solutions? This is beyond the scope of my question (and I'm taking a PDE class next semester), so feel free to skip it.

2. On a related note, is there a proof or heuristic explanation for why the given boundary conditions leave that particular degree of freedom? How might I go about proving that a given solution is unique up to the value of the circulation?

3. The essence of my question was: "if the freestream velocity and constraint on the normal derivative to the cylinder aren't enough to determine the unique solution, then how do we find the circulation and thence the lift on the body analytically? Anderson introduces the lifting and non-lifting cylinder, mentions the Kutta condition in a rather abstract way, and then starts trying to satisfy the boundary condition and a constraint on the vortex strength at the trailing end using a vortex sheet. His assertion seems to be that if we can construct a vortex sheet that satisfies these conditions (i.e. the boundary of the aerofoil is a streamline with the vortex intensity of the trailing edge zero), then the circulation induced by this sheet will accurately give the lift/circulation, but it isn't clear why this would be true. I have some time to work through this tonight, so I may be able to answer my own question.

Please feel free to answer as much or as little of this as you want. I suspect I can also puzzle some of this out on my own. Thank you very much!

1. The Laplace equation for the potential has unique solutions (up to an irrelevant constant) only when boundary conditions are given everywhere on the boundary of the domain. In your case, such conditions were only given on the cylinder surface. In this case there is an infinite family of possible solutions, distinguished from each other by the value of the circulation. The uniqueness theorem does hold for multiply connected domains also.
2. Anderson's language seems to be a bit sloppy here (I don't have the book in front of me, so I don't know all the context; one also has to keep in mind his intended audience, which limits the amount of rigor that's appropriate here). Long story short, I think what the author was trying to say is that we can think of the flow in question as being generated by a rotating cylinder in viscous flow. Notice that the potential flow he is discussing is indeed a solution of the full Navier-Stokes equations, since the viscous term drops out for irrotational flow. Thus, if we consider a two-dimensional cylinder in a viscous fluid within an unbounded domain, which starts spinning around its axis at $t=0$, then there is a Navier-Stokes solution that asymptotically converges to the potential flow solution with circulation for $t\rightarrow\infty$.
3. I am not entirely sure I understand your question correctly, but it seems to be related to the first item regarding boundary conditions and uniqueness. The reason to use vortex sheets (or potential vortices) is that we cannot generate a potential flow with circulation without these, at least in the discrete case. It is, however, possible to construct doublet density distributions that result in non-zero circulation.

1. Yes, the boundary condition of $\mathbf V\rightarrow\mathbf V_\infty$ for $x, y\rightarrow\infty$ (notice the typo(s) in your question) is not sufficient.

2. Perhaps the easiest way to see this is to examine your example of the cylinder flow: The flow without circulation has velocity vectors that are tangential everywhere to a circle with radius $R=\sqrt{\kappa/V_\infty}$. Now you add the term $\Gamma/2\pi\,\,\theta$ to the potential. This is the so-called potential vortex, corresponding to a velocity field that has circular streamlines, and the velocity vectors are tangential to circles around the origin everywhere. It is obvious that the resulting velocity field will therefore still be tangential to the above circle with radius $R$. Notice also that the velocity field of the potential vortex decays as $1/r$, so your boundary conditions at infinity remain satisfied also. Thus, there is an infinite family of solutions to the potential flow problem around a cylinder, distinguished by the value of the circulation $\Gamma$. Now, to get away from the simple geometry of the circle, perhaps the easiest path is to realize that a conformal mapping of the plane takes a solution to the Laplace equation to another solution of the Laplace equation in the transformed plane. Cutting things short a bit, it turns out that we can now map our disk with radius $R$ to any singly connected shape we want, including the shape of an airfoil. Good examples of transformations that do this are the Joukowski transform, and the Karman-Trefftz transform.

3. Perhaps the most intuitive way to see the role of the Kutta condition is as follows: Just as for the circle, there is an infinite family of potential flow solutions for the flow around an airfoil. Almost all of these solutions have a rear stagnation point that lies somewhere on either the upper or lower surface of the airfoil. Such flows are not physical, since they would require fluid to move around the (almost) sharp trailing edge of the airfoil. That's not possible because it would require infinite pressure gradients. The Kutta condition simply says that, for a real airfoil, the circulation will always be such that the rear stagnation point is located at the trailing edge.

• Thank you very much for your help/time. I've been trying to teach myself aerodynamics/CFD over break, and it's great to have help from someone knowledgeable. I have a few quick follow-up questions at your convenience. They're too long for a comment, so I've edited the initial post. Thanks! Jan 1 '17 at 21:53
• @Pirx, very nice answer. I have used Anderson's book in my past undergraduate aerodynamics classes, but have always been aware of his lack of rigor. But you are correct, most of his intended audience wouldn't even notice the lacking rigor as they are engineering students, ha!
– TRF
Jan 1 '17 at 23:52
• These are addressing two different kinds of situations. Using conformal mappings (Joukowski or Karman-Trefftz for specific airfoil shapes, or Schwarz-Christoffel transforms to approximate arbitrary airfol shapes via polygons, for example) you can generate (approximate) potential flow solutions for pretty much any airfoil shape. On the other hand, you can approximate airfoils very roughly as just a camber line, so you have an infinitely thin airfoil. Anderson must explain somewhere why circulation density distributions are the appropriate vehicles to achieve this (if not, feel free to ask).
– Pirx
Jan 3 '17 at 0:03
• Contd. The big benefit of this approach is that you get approximate closed-form solutions after using the Glauert transform. This is mostly useful to elucidate some fundamental effects of airfoil geometry on lift and moments generated by thin airfoils.
– Pirx
Jan 3 '17 at 0:06
• I forgot to add, nowadays people mostly like to use panel methods based on piecewise polynomial representations of singularity distributions (of source, doublet and/or vortex densities) and conformal mappings are considered a bit old-fashioned, useful mostly for derivations and theoretical work. This is perfectly fine when one is interested in finding velocity and pressure distributions for given airfoil shapes. However, for the "design problem", people specify desired pressure distributions and then want to find the corresponding airfoil shape. Conformal mappings are attractive in such cases.
– Pirx
Jan 3 '17 at 0:19

I don't have the book in front of me, but the essential point is that for must "real world" flow situations, the fluid viscosity is very relevant in formulating the boundary conditions (i.e. there is no "slip" between the tangential velocity of the boundary and a viscous fluid), but its effect on the solution in away from the boundary is often negligible.

Therefore, a good approximation to the flow pattern round a rotating cylinder can derived by ignoring the fluid viscosity completely (which is equivalent to using the Laplace equation and a solution method using potential flows, instead of the full Navier-Stokes equations), except in specifying the boundary conditions.

This is indeed related to the Kutta condition, which is a way to capture "the global effect of viscosity on the flow close to an aerofoil" without actually solving for the viscous flow field. Physically, if there was a rapid change between the flow velocities over the pressure and suction surfaces at the trailing edge, viscous effects would cause that discontinuity to dissipate through the fluid at the speed of sound, and therefore travel upstream around the aerofoil since we are considering subsonic flows.

Idealizing the physical situation, the Kutta condition says "let's ignore the viscosity, but impose the condition that there is no singularity in the flow field at the leading edge". We can then calculate the irrotational and circulatory flow fields separately, and add them together since Laplace's equation is linear.

The resulting flow field does not satisfy the real-world boundary conditions on the aerofoil, but unless the flow separates from the aerofoil (which is also caused by the fluid viscosity!) it is a reasonable approximation to the flow outside the boundary layer, and gives a reasonable approximation to the lift force - but not the drag force, which also depends on how viscosity effects the flow in the boundary layer.