For centuries people have obtained qualitative and quantitative information about various

flow fields by observing the motion of objects or particles in a flow. For example, the

swirling motion of a tornado can be observed by following debris moving within the

tornado funnel cloud. Broadly speaking a tornado is a rotating mass of air circulating

around a low pressure central core. Historically it has been difficult to get comprehensive

experimental data throughout a tornado or hurricane, thus, detailed theoretical techniques

and computational fluid mechanics are helping to shine some light on the complex fluid

dynamics involved in such a flow. The motivation for this project is to develop a

preliminary model to predict pressure and velocity distribution within and around a

tornado.

In lecture # 4, we approximated a tornado flow with a

free vortex model with circular streamlines. The

velocity field was obtained from continuity, and then

pressure was obtained from the momentum equation

which also agreed with Bernoulli equation. The model

predicted the flow pressure decreases as we approach

the origin (the eye of the storm). This is consistent

with experimental measurements where they show the

pressure outside a tornado is typical atmospheric

pressure which is larger than it is near the center of the tornado where an often

dangerously low-partial vacuum may occur ingesting 1-mm-diameter sand from the

surface as it encounters a debris field.

Despite its interesting results, this simple model cannot be used to approximate the

tornado flow throughout the flow field! Why? The free vortex model cannot be usedÂ

because the origin itself is a singularity point and the pressure and velocity become

infinity at the center. To avoid the singularity there and a more physically realistic model

of a tornado, it is suggested to develop a two-region model when the outer region (e.g.

r>R) is modeled with a free vortex flow and the inner or core region (0

by solid body rotation. In this new model, the flow in the inner region is still inviscid, but

rotational.Â

Â

Let us consider a horizontal slice through a tornado. The flow in this horizontal plane has

no dependence on elevation z. Consequently, the effects of gravity are negligible. Within

this particular horizontal slice it is assumed that the flows are steady, inviscid, and

incompressible with streamlines in the horizontal plane.Â

First we focus on the inner region fluid (0

(solid body rotation) around the z-axis. The steady incompressible velocity field is given

by ur=0 utheta=rw uz= 0. The pressure at the origin is equal to P intial. Using theappropriate component of momentum equation and neglecting gravity, obtain anÂ

expression for the pressure filed in this region (0

Ï‰, Ï, Pinitial). Your solution should clearly show that P increases parabolically with Â r.Â

Â

(b) Results obtained in part (a) may sound conflict with the Bernoulli equation! How

come both pressure and velocity increase with r while the Bernoulli equation predicts

higher speed for a lower pressure region? To answer it, we should investigate whether the

Bernoulli equation is still valid in the inner region or not. To start, note that our flow

condition satisfies the Euler equation:

(equation in attachment)

We are interested to obtain an alternative version of the Euler equation. To do that, first

show the following vector velocity in Cartesian coordinates for a three-dimensional flow: (equation in attachment) Â

Where V is the magnitude of the velocity vector.Â For full credit, expand each term asÂ

far as possible and show all your work.

Show that the Euler equation can be then rewritten into a form showing that the

gradient of three scalar terms is equal to the velocity vector crossed with the vorticityÂ

vector :

(equation in attachment)

Where g represents gravity acting in the z-direction and z is vertically upward. This

formula shows for an irrotational flow, i.e. vorticity vector=0 , the scalar Â P/Ï +V^2/2 + gz Â must be aconstant value at every point in the flow filed. It can be argued that even for a rotational flow where vorticty vector not = 0, the scalar Â p/Ï +V^2/2 + gz must be constant along a streamline. In other words:Â

(equation in attachment)

Therefore the Bernoulli â€œconstantâ€ C is constant only along a streamline for a rotational

flow; but changes from streamline to streamline. This is typical of rotating flow fields.

Â

Subject | Science |

Due By (Pacific Time) | 04/30/2015 11:00 am |

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