Vortex are found in many shapes and sizes in nature but all of them have some similarities.So I will go back and forth between comparing vortex generated in air and in liquids. Which is valid because in fluid mechanics both gases and liquids are categorized together as Fluids.
That is why submarine models can be tested in wind tunnel and aeroplane models can be tested in water tunnels.
2 Dimensional analysis of a vortex shape:
Orthographic Top View Projection :
The Orthographic Top View Projection of a vortex can be seen in two different ways.
If the viewer is interested in the analysis of the spirals created by a vortex then the viewer can consider the shape to be spiral (which actually is the mathematically correct definition for the shape of the vortex from top view.)
Hence the mathematically correct shape for the orthographic top view projection of a vortex if the viewer is interested in the spirals created in the vortex will be a spiral.
On the other hand if the viewer is not interested in the spirals but just
the primitive shape that is created then the shape can be stated to be an ellipse .
Many people may disagree by saying,"What do you mean ellipse ? The shape is a circle!"
Well they are partially correct.
Because a circle in-fact is a special case of an ellipse where the foci are situated on top of each other, which makes the focus of the circle.
But it will be mathematically correct to state the shape as an ellipse because , due to the transient and turbulent nature of fluid flow that make the vortex , the shape will be asymmetrical and hence it can never be a perfect circle.
Hence the mathematically correct shape for the orthographic top view projection of a vortex if the viewer is not interested in the spirals created in the vortex will be an ellipse.
3 Dimensional analysis of a vortex shape
Note: This section still under investigation , further information will be uploaded upon completion.
The shape of the whirpool are axially asymmetrical concave paraboloid.
However the axis of the whirlpool or tornado can be a curve. Which gives us the familiar curved look of a tornado .
One example of axially symmetrical concave paraboloid is Rotating furnace https://en.wikipedia.org/wiki/Rotating_furnace
Whirpools become assymetric because of high reynold's number creating turbulence in the flow of the fluid.
Also it is noted that:
Different vortices have different kinds of shapes associated with them.
For example some vortex are torroidal like smoke rings one blows after taking a puff of cigarette.
.They are called vortex rings