Simulation of Tornados
[CBPrice 24-03-13. Draft1]
[CBPrice 08-04-13. Rev1]
Introduction
There are a number of models for tornados around. Most seem to focus on modelling the velocity of the air
movement within the tornado. A good summary of recent models is found in the paper (Kilty) presented on
the module web-pages.
These models are interesting, since they do not start with the forces on objects within the reach of the
tornado, but they start with the velocity of the air and therefore of the objects. Most of our simulations have
employed the workflow (i) identify the forces, (ii) calculate the accelerations, (iii) update the velocities and
finally (iv) update the positions. But now we lose the first two stages and start with equations for velocities.
So our workflow is (i) use the velocities to update the positions.
Let’s have a look at the velocities we need, there are three. First there is the radial velocity which is the
velocity of the movement of objects towards the eye of the tornado. Then there is the azimuthal velocity
which is the velocity at a tangent to the movement of the object around the eye. Finally there is the vertical
velocity which is the upward velocity of the objects due to the suction of the tornado. These are illustrated in
the figure below. On the left we see a 3D view showing the movement of an object picked up in a tornado.
The centre two diagrams show the radial and azimuthal velocities (looking downwards on the tornado) and
the right diagram (a side view) shows the direction of the vertical velocity.
Burgers-Rott Model
This model specifies three equations for the velocities within a tornado. They are analytical solutions of the
“Navier-Stokes equations” (from the theory of fluid dynamics) which specify the radial, azimuthal and
vertical velocities.
𝑉𝑟 (𝑟) = −𝑎𝑟
Γ
2
𝑉𝑎𝑧 (𝑟) =
(1 − 𝑒 −𝑎𝑟 ⁄2𝜈 )
2𝜋𝑟
𝑉𝑧 (𝑧) = 2𝑎𝑧
Here a is the suction coefficient, which describes how fast the object move towards the tornado eye, and also
how fast they are sucked upwards. Also Γ is the circulation coefficient which determines how fast the
tornado air is circulating around the tornado eye. Finally 𝜈 is the viscosity of the air, which is a measure of
damping or energy loss. Think of stirring a cup of tea, and think of stirring a cup of treacle. The latter is
more viscous. What is the difference in your stirring?
Initial Simulations
Two sets of coefficients may be useful. (i) Γ=5000, a = 0.004, 𝜈 = 5. See the Kilty paper with circulation
reduced a bit. The single debris object was located at an initial radius 60. Here are plots of orbit, radius and
height versus time
Here is a simulation with the debris located at an initial radius 15
(ii) (i) Γ=1000, a = 0.04, 𝜈 = 5
These figures show the importance of choosing the correct parameter set for student appreciation of the
model. While both simulations show an inward spiral to the tornado centre, the first show clearly that the
rate of spiralling inwards slows down and almost converges. Note that also in the second simulation the
debris has exited the level (5000m) which will cause UDK to exit (crash).
Current Code and Levels
CBP74_Tornado Class
Consists of a tornado centre with one element of debris within a single class, both as components. Allows
logging of the debris behaviour. This is a toy example and is now deprecated. Do not use!
CBP_TornadoDebris and CBP74_TornadoCentre Classes
The CBP74_TornadoDebris class contains the code required to update each debris element. Typically many
instances of this class will be placed in a level. Each can be assigned its own static mesh. The
computation(dT) function is called by the CBP74_TornadoCentre computational loop. Each actor of this
class searches for the CBP74_TornadoCentre class to discover the tornado centre.
The CBP74_TornadoCentre class manages the debris elements. It sets their properties (suction, circulation,
viscocity) on initialisation and allows these to be changed from the command line for all debris elements.
This class contains the computational loop and displays the time as well as the common parameters.
Example Levels
CBP74_TornadoDebrisTest1.udk contains a random collection or debris elements.
The parameters are
Γ= 5000
a=0.004
𝜈=5
CBP74_TornadoDebrisTest2.udk contains a specific collection of debris elements.
The parameters are
Γ= 5000
a=0.04
𝜈=5
which gives a radius at maximum tangential (azimuthal) velocity of 1.17m (930UUs). Debris elements are
located at 256, 512, 768, 1024, 1280, 1536 to span this range. Here’s a screenshot of the ICs and developing
scenario
and here’s the plot of tangential and (negative radial velocities) versus radius from Matlab
35
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5
0
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CBP74_TornadoDebrisTest3.udk contains a specific collection of debris elements.
The parameters are
Γ= 5000
a=0.04
𝜈=5
As above in test2 but the heights of the elements are all different.
Top left init. Bottom left after 6 secs. Right later. Nice tornado.