# Is there any solution to the potential flow around a square cylinder?

Potential flow around a circular cylinder is a classic solution. But I am wondering if there is any solution similar to this for the flow past a square cylinder?

• Removed the turbulence and navier-stokes since, by definition, potential flow is inviscid and irrotational and so both tags don't apply. I added the potential-flow since it is substantially different from existing tags. – tpg2114 Dec 13 '13 at 21:30
• Thanks, I used those two tags just in hope that other "turbulence and navier-stokes" guys may see this post and share some ideas too. Cause I am afraid the fluid people is not too many here. – Daniel Dec 13 '13 at 21:33
• There's not many, but I'm one :) Unfortunately I don't think there is a solution -- I answered below. – tpg2114 Dec 13 '13 at 21:36
• While - as tpg2114 notes - the problem is not physically realizable because the irrotational and inviscid assumptions for it are never realized, the mathematical problem can indeed be solved, by using a Schwarz-Christoffel conformal transformation to map the square and its outside to opposing half-planes. – Emilio Pisanty Dec 14 '13 at 2:13

Flow around a square cylinder

Similar solution to a cylinder but more violent von-karman vortex street because separation happens at an edge..

Notation of variables for a flow around a square cylinder The Reynolds number is defined as $Re = Ud/\nu$ and stands for a ratio between the inertial and viscous forces.

a) Re<55

Re=30, alpha=0 Re=30, alpha=0 a) Re>55

Karman vortex street: Re=30, alpha=0 Karman vortex street: Re=30, alpha=0

• Any analytical results? Or references that I can look into? Thanks – Daniel Dec 13 '13 at 21:27
• If this is your data, can you post a movie of these on youtube and link it here? I'd love to see them. – Kyle Kanos Dec 14 '13 at 3:44
• What program are you using to simulate the dynamics. – unsym Dec 17 '13 at 8:49
• This is done in openFOAM. Very simple to learn and start coding in 1 day. Check it out. – mcodesmart Dec 17 '13 at 21:57

I am not an expert, but the thing I would do would be to use conformal invariance of potential flow. You would find a conformal transformation take a circle into a square, then take your potential function for the circular cylinder, and put this function through the conformal transformation. Derivatives give you the velocities for the square geometry.

I am not sure exactly what the conformal transformation is, but it won't be something you can do analytically because it involves elliptic integrals (I think). Also, the solution will probably have singularities at the corners of the square, judging by what other people are saying. Therefore the solution found this way won't be physically realizable.

• +1 The name you're groping for is Schwarz-Christoffel mapping and it will certainly work in principle. You're likely right about elliptic-like integrals, but sometimes you can do the integrals in closed form. Mathematica is rather good at Schwarz-Cristoffel integrals. – WetSavannaAnimal aka Rod Vance Dec 14 '13 at 2:22
• Or maybe it can be done analytically since you only care about derivatives of the map, so the integral in the definition of the mapping goes away. It's something to look at, at least. And if you use the map you link to, you still need to compose it with a transformation taking the upper half plane to a disk. Of course a simple Moebius transformation will do. – Brian Moths Dec 14 '13 at 2:28
• I wonder if you could specify the solution at the singular points like you would on an airfoil and set the velocity to 0 akin to the Kutta-Condition. It's been a really long time since I've done conformal mapping and it was never with squares... – tpg2114 Dec 14 '13 at 16:18

The flow around a square is dominated entirely by viscous effects and the vortex shedding due to the boundary layer. Additionally, at very high Reynolds numbers such that viscous effects are minimal, the square has considerable separation which cannot be solved with the potential equations.

Because potential flow requires both irrotational and inviscid assumptions, I do not think that it can be used for the flow around a square cylinder. It works well for circular cylinders only at low enough Reynolds numbers that the flow does not separate around the cylinder. But for the square, that doesn't hold.

• Okay, you are right. Do you know any viscous solution to the flow past a square cylinder, or any related paper? I am very interested in these type of sharp cornered external flow. – Daniel Dec 13 '13 at 21:41
• My only experience with square cylinders is to validate the Large Eddy Simulation solver we use in our lab. This search will turn up more papers than I can summarize that might help you find what you need. – tpg2114 Dec 13 '13 at 21:44
• I am well aware of these simulations and papers. I run these simulations myself. These days, I am thinking of force history issue. As you refine the mesh or as you choose different turb models, the unsteady force turns out to be different, this bothers me. So I want to start from a potential flow results. I use a potential solver to get an initial field. But I found as the mesh is refined, the velocity at the corner cells keep increasing. – Daniel Dec 13 '13 at 21:51
• @Daniel It's because it's ill-posed in potential flow unfortunately. Mesh refinement in LES is pointless really unless you are explicitly filtering the fields. But implicitly filtered LES, forget about it. We can talk in chat if you'd like to discuss more about it -- I have these arguments with my advisor all the time :/ – tpg2114 Dec 13 '13 at 22:05