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Introduction and Background

\(^{\circ}\) Fluid flow and patterns have long been studied in fluid dynamics. In particular, von Karman vortex streets have been an area of great interest. This striking phenomenon has been a pertinent consideration across many fields, and is an excellent example of flow behavior.

Von Karman Vortex Streets are flow patterns generated in the wake of flow past an obstacle (generally a cylindrical object is used as an obstacle). Named after Hungarian aerospace engineer and physicist, Theodore von Karman, these “streets” are results of unstable flow. They are characterized by periodic vortex shedding where alternating vortices form in flow parallel layers past the obstacle. The formation of vortex streets is dependent on the Reynolds number, which will be further discussed later.

This phenomena is present in many settings: on a large scale, atmospheric flow can be disrupted by islands or mountains, causing noticeable and powerful vortex streets downwind; on a smaller scale, tall buildings also disrupt wind flow, causing cross winds and eddies in urban settings, and turbulence in river flow plays an integral part in sediment displacement. Therefore the study of von Karman Vortex Streets is relevant in fields such as meteorology, aviation, environmental studies, and engineering. In this thesis, we will explore Karman vortex streets, and seek to create a practical laboratory demonstration of vortex street formation.

Definitions and Derivations:

In this section we define and derive key terms in order to study von Karman streets. In the interest of defining a vortex, we initially seek to define circulation and vorticity: circulation can be defined as the line integral of the velocity field about a closed loop, \(S\): \[\Gamma=\oint_S \left( V \cdot dl\right)\]

Applying Stokes’ Theorem, we can then relate the circulation to the vorticity: \[\Gamma=\oint_{\delta S} \left(V \cdot dl\right) = \int \int_S \left (\omega \cdot dS\right)\]

A vortex is characterized by its vorticity, which describes the local rotation at a point in the fluid. The vorticity of a fluid material is given by: \[\boldsymbol\omega = \nabla \times \textbf{u}.\] In other words the vorticity is the curl of the velocity vector field. We can see that it will be a scalar quantity in two dimensions. We can also observe that it is possible to have circulation and vorticity in a vector field without observing vortices, such as in shear flow. Therefore it is important to clarify the definition of a vortex as a compact circulating region of fluid (i.e. a localized piece of vorticity). We can use a general normalized Gaussian function to represent vorticity that takes the form
\[\boldsymbol\zeta\textbf{(r)} = \frac{e^{-r^2/\epsilon^2}}{\pi\epsilon^2}\] where represents the “width” and determines how fast the function will decay. Therefore, we see that in two dimensions, the Gaussian contains a localized region where it reaches its peak, and then exponentially decays away as we move further away from this peak. If we take the _0, this then becomes a delta function where we have infinitely large peak and zero everywhere else. This would thus be the case of a point vortex, which we can think of as a two dimensional Dirac delta function.

Vortex Shedding is described as the process by which a vortex street is formed (Tritton 1988). This is a phenomenon in which flow is disturbed by an obstruction such that attached vortices are periodically “shed”. These vortices produced by shedding are also called eddies. If we think of fluid as moving in