CHAPTER 15 PROJECT Modeling a Tornado

A tornado is a three-dimensional wind field that can be modeled as a velocity field. In this project we look at a two-dimensional slice of a vortex and use an Oseen vortex velocity field to model a tornado.

In rectangular coordinates, an Oseen velocity field is given by \[ \mathbf{v}=\left(\dfrac{V_{\max }\,y}{\sqrt{x^{2}+y^{2}}}\right)\mathbf{i}-\left(\dfrac{V_{\max }\,x}{\sqrt{x^{2}+y^{2}}}\right)\mathbf{j} \]

We place the center of the tornado at origin, where \(\mathbf{v}\) is undefined. Then \(\mathbf{v}\) gives the velocity of the wind at any other location within the vortex.

1. Show that in the Oseen model, the speed (magnitude of the velocity) of the wind at any location is \(V_{\max }\).
2. Suppose \(V_{\max }=80{m}/{\!s}\). Describe the velocity field at the following locations \[ \hbox{(a) (0,-10), (b) \((-10,0)\), (c) \((0,10)\), (d) \((10,0)\), (e) \((10,10)\)} \]
3. Add a few more vectors in the velocity field. What can you conclude about the direction of the wind?

   A fluid, like the air in a tornado, is considered to be incompressible if the divergence of its velocity field, \({div}\mathbf{v }\), equals 0. When a fluid is incompressible, it is possible to find a function, \(\psi ( x,y) \) , called a stream function. The level curves of a stream function are called stream lines. These are the curves along which particles of fluid would flow. (You may have seen examples of stream lines such as air moving around a car in a wind tunnel.) The stream function, \(\psi ( x,y) \), for a velocity field, \( \mathbf{v}=h( x,y) \,\mathbf{i}+g( x,y) \,\mathbf{j}\), where \(h\) and \(g\) are functions of \(x\) and \(y\), satisfies \[ \dfrac{\partial \psi }{\partial y}=h( x,y) \quad\hbox{and}\quad \dfrac{\partial \psi }{\partial x}=-g( x,y) \]

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4. Show that in the Oseen model, the air in a tornado is incompressible.
5. Find a general form for the stream function of a stream line \( \psi ( x,y) =C\) for an Oseen velocity field.
6. Suppose the speed \(V_{\max }\) of the wind in a tornado is \(80 \rm{m}/{\!s}\) and the value of the constant \(C\) is \(3200 \rm{m}^{3}/{\!s}\) (the units of \(C\) represent a volume flow rate).
   1. Find the stream function and graph the stream line \(\psi (x,y)=3200\).
   2. Graph the stream lines for \(C=1000,\) \(2000,\) \(3000,\) and \(4000\).
   The air in the tornado is turning, and the twisting behavior of the velocity field can be quantified by a vector \(\mathbf{w}\), called the vorticity. The vorticity at a point in space is a vector that is normal to the plane of rotation, and describes the amount of rotation about that point. The vorticity of a vector field is given by the curl of the vector field, curl \(\mathbf{v}\). That is, \[ \mathbf{w}=\rm{curl}~\mathbf{v} \]
7. Find the vorticity \(\mathbf{w}\) of the Oseen velocity field. Interpret the result.
8. In this project, we assumed the wind speed in the vortex was \(80 \rm{m}/{s}\). Research the Enhanced Fujita scale, or EF-scale, that is used to classify tornados based on the damage they cause. Rework Problems 2, 3, 6, and 7 for a tornado with a wind speed of \(110 \rm{m}/{s}\) (an EF-5 tornado). Compare the results to a tornado with a wind speed of \(80 \rm{m}/{s}\).