# 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 layers in a laminar flow, when this laminar flows comes in contact with an obstacle, the “layers” that are in contact with the body lose speed. Due to the viscosity of the fluid (the transfer of momentum between layers within the fluid), shear arises between the layers, as those moving further away from the body are moving quicker than the obstructed ones. Von Karman vortex streets are characterized by periodic and symmetric vortex shedding, and the appearance and behavior such shedding is predicted by the Reynolds number.

That is, when a fluid particle approaches the edge of an obstruction (say, a cylinder) an increase of pressure acting on the particle due to the slowing down of flow near the obstacle, becomes disruptive to its movement. At the leading edge of the cylinder the high pressure drives the fluid flow about the object, developing boundary layers. Once at the widest diameter of the cylinder, these boundary layers separate from the edges of the object and form shear layers where different “layers” of the fluid are flowing at different speeds, thus producing shear between each. These shear layers trail in the flow of the fluid layers surrounding it. That is, the layers actually in contact with the cylinder move much slower relative to the layers furthest away and in contact with the free flow. If the difference in speed between shear layers is great enough, the shear created will cause the slower layers then roll into the near wake past the object (towards the the bottom edge of the cylinder) and consequently fold into each other, merging into discrete vortices. Once enough pressure builds, the circulating vortex release from the area at the base of the cylinder and proceed in the direction of flow. This process alternates about the opposite sides of the cylinder and thus forms a periodic and symmetric pattern trailing downstream of the obstruction (NASA 2012).]

Therefore, we can apply this understanding to the physical phenomena mentioned in the introduction: Most clearly, we can see these flow patterns in streams where water flows past rocks. These vortex streets that are created can carry sediment in a particular manner, which then becomes important to geologists and environmental scientists. On a grand scale, tall buildings, mountains and islands can be examples of bluff bodies that disrupt airflow and create less visual but very powerful vortex streets that affect fields from aviation to meteorology.

As mentioned previously, the Reynolds number is a parameter used to classify flow behavior in fluid dynamics. This dimensionless parameter is described as the ratio of inertial forces to viscous forces, and is given by:

$Re = \frac{\rho vL}{\mu} = \frac{vL}{\nu}\\$

where $$\rho$$ is the fluid density $$v$$ is the mean flow speed, $$L$$ is the characteristic linear dimension (e.g. hydraulic diameter, traveled length of fluid), $$\mu$$ is the dynamic viscosity, $$\nu$$ is the kinematic viscosity ($$\nu = \frac{\mu}{\rho}$$).

Typically, low Reynolds numbers results in laminar flow. However as $$R$$e increases and reaches a critical value of about $$Re\simeq 40$$, the flow becomes unstable and characteristic flow patterns appear in the flow past an obstacle. This instability becomes more apparent further downstream and thus gives rise to vortex or eddy shedding. When the Reynolds number exceeds 400, we begin to see turbulent flow, in which von Karman vortex streets disappear. [Figure 1]

Viscosity describes the internal friction of the material, or in more general terms, the resistance or opposition to flow (Online 2014). It is the ratio of the shearing stress to the velocity gradient and is often called the dynamic viscosity. The kinematic viscosity is then the ratio of the dynamic viscosity to the fluid density. Since these quantities are intrinsic characteristics of a fluid, they are specific to each problem.

Similar to the Reynolds number, the Strouhal Number is a dimensionless quantity that describes oscillating flow mechanisms. It is often used to give the ratio or relationship between flow rates and frequency. Labelled St, it is given by: $St= \frac{nL}{u_0}$
where $$n$$ is the frequency with which the vortices are shed in the wake of the obstacle, $$L$$ is the characteristic length, and $$u_0$$, the fluid velocity.

We see that for large Strouhal numbers (i.e. on the order of $$1$$), viscosity will dominate fluid flow. Consequently we begin to see collective oscillation of the fluid “plug” - i.e. a large vortex. Alternatively, smaller Strouhal numbers ($$\leqq 10^{-4}$$), the high speed of the fluid will dominate flow. This get characterized by a quasi steady state flow. Thus at intermediate Strouhal numbers, we see the buildup and shedding of vortices in the flow (Taylor 2003, Sobey 1982).

## Mathematics

It is now necessary to discuss the governing equations of fluid motion. The equations arise from the fundamental concepts of conservation of mass and momentum. We first consider mass or volume conservation. If one imagines a cube or region of arbitrary volume, $$V$$, where fluid moves in and out at all points of its surface, we can describe the rate of the decrease in mass as $V=-\frac{d}{dt} \int\left(\rho dV\right) = \int\left(\frac{\partial \rho}{\partial t}\right) dV,$ where $$\rho$$ is the fluid density. Mass must be conserved, therefore the decrease of mass must also equal the rate of flux out of the volume region $$V$$. The rate of loss of mass from $$V$$ is then: $\phi = \int_S \left(\rho \textit{u} \cdot dS\right)$ where $$dS$$ is an element of $$S$$, the surface of the volume, and $$u$$ is the velocity at the velocity at $$dS$$.

Physically, we know that the part of the velocity vector perpendicular to the surface produces flux or flow out of the object. Using Green’s formula, this can also be written as: $$\int_V(\nabla \cdot \rho u) dV$$.

Since we are interested in mass at a point (instead of volume), we want to now consider an infinitesimally small volume and take: $\frac{\partial \rho}{\partial t} = -\lim_{x \to 0} \int\left(\rho \it{u} \cdot \frac{dS}{V}\right) \Rightarrow \frac{\partial \rho}{\partial t}= -\nabla \cdot \rho \textit{u}$ This defines the conservation of mass in any fluid of volume $$V$$. Rearranged, this gives: $\frac{\partial \rho}{ \partial t} + \nabla \cdot \rho u = 0$
This relationship is known as the continuity equation and will be the basis from which we will derive further equations of motion for fluids.

### Euler’s Equation

Euler’s equation for (inviscid) fluid motion follow from Newton’s second law, $$F = ma$$. Applying this equation to a unit volume of fluid, the force $$F$$ on the fluid can be described as $F= -\nabla p \cdot dV,$ where $$\nabla p$$ is the gradient of the pressure field and $$V$$ is the volume. Still, $$F = m a$$, or $$F = m dv/dt$$. Combining these equations gives: $-\nabla p V = \rho V \frac{D u}{ Dt } ,$ where the mass is given by the density, $$\rho$$, multiplied by the volume and the velocity of the mass is given by applying the substantive derivative $\frac{D}{Dt} = \frac{\partial}{\partial t} + u \cdot \nabla$ to the velocity vector (such that the velocity components of the fluid are expressed in terms of the field).

Simplifying this equation: $-\nabla p = \rho ({ \frac{\partial u}{ \partial t} + u\cdot \nabla u})$ We can then rearrange this to get $\frac{\partial u}{\partial t} + (u \cdot \nabla) u = \frac{-1}{\rho} \nabla p$ This gives Euler’s equation for inviscid flow.

Alternatively, this may also be written as:
$\\ \rho\frac{ D\textbf{u}}{ Dt} = \rho(\frac{\partial{\textbf{u}}}{\partial t} + \textbf{u} \cdot \nabla \textbf{u})$
This tends to be a more common representation of Euler’s equation. (Denker 2008)

### Navier Stokes equation

The Navier-Stokes equation is derived similarly to the Continuity and Euler equations. As shown above, Euler’s equation for inviscid flow is a clear manifestation of Newton’s second law, with the left-hand side representing Newton’s second law ($$\rho\frac{D\textbf{u}}{Dt}$$ ) and the right-hand side, the sum of the forces. Since real fluids are never truly inviscid, we must now consider the nature of these viscous forces (see (Tritton 1988), 5.6 pp. 52 for further details).

Firstly, we have external forces acting on the fluid and are thus defined and particular to the specific problem. These external forces acting on the body of fluid will be denoted as $$f$$. Next, we consider forces due to the pressure and viscous action, which are intrinsic qualities in the dynamical equation. Both viscous forces and pressure create internal stresses in the fluid. That is: the force on a fluid particle is the net effect of the stresses over its surface (see (Tritton 1988), 5.6 pp.52).

In considering the force due to pressure, the net force per unit volume is simply $$\frac{\partial p}{\partial x}$$ (in the $$x$$ direction). To consider all directions, this becomes -$$\nabla p$$, for a general pressure field. The viscous term, however, is more subtle. Since viscous stresses oppose the motion of fluid particles relative to one another, or the deformation of these particles, they are dependent on the rate of deformation (and consequently the velocity field) and the properties of the fluid (i.e. viscosity). Assuming the fluid has constant density, the viscous force per unit volume becomes $$\tau = \mu\nabla^2u$$ , where $$\mu$$ is the coefficient of viscosity and u is the velocity field.

In the general case, the stress is a second order tensor, $$\sigma_{\textit{ij}}$$, a quantity with magnitude and two directions: $$i$$ being the component of stress on a surface element $$\delta S$$. and $$j$$, the direction of the unit normal $$n$$. Thus, the total stresses in a fluid can be defined as: $\sigma_{\textit{ij}}= -p \delta_{\textit{ij}} + \tau_{\textit{ij}}$

Combining all contributions, we can create the full expression: $\rho\frac{\partial D\textbf{u}}{\partial Dt} = -\nabla p + \mu\nabla^2u + \textit{f}$ This is known as the Navier-Stokes Equation. We can again apply the substantive derivative and continue assuming constant density, thus obtaini