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# An introduction to vortex flows - theoretical hydrodynamic vortex behaviour

12 October 2016

## Theoretical hydrodynamic vortex behaviour

Due to the difficulties in defining vortex behaviour it has become common to identify quantitative features associated with vortices.

As a starting point it is helpful to distinguish between the behaviour of the vortex core for real fluids and ideal fluids (a fluid with no viscosity), as this is where the most significant differences are evident.

### Forced vortex behaviour

For a real fluid the vortex core is defined as the region of the flow field where solid body rotational or forced vortex behaviour is dominant. This behaviour occurs due to the viscous forces in the fluid becoming significant compared to the inertial forces.

The local Reynolds number is the ratio of the inertial force and viscous force in the fluid. A low Reynolds number indicates a relatively larger viscous force, signifying the tendency to behave more like a solid body.

Low Reynolds number flows will therefore tend to exhibit a forced vortex behaviour i.e. at the vortex core. As the vortex behaves as a solid body in a forced vortex the angular velocity about the axis remains constant. Therefore the tangential velocity increases linearly with increasing radius:

Uθ=Ωr

where Uθ is the tangential velocity, Ω is the angular velocity and r is the radius.

Any fluid parcel in the flow is rotated as it is travels round the vortex, as shown in Figure 1.

Figure 1: Forced vortex behaviour

For this reason a forced vortex is also called a rotational vortex.

### Free vortex behaviour

When the Reynolds number of the flow is large the effect of the viscosity of the fluid becomes small.

Taking this behaviour to its limit, the effect of the viscosity becomes negligible and the flow behaves in an inviscid manner (with no viscosity). This results in another behaviour, normally called a free vortex.

In a free vortex the angular momentum remains constant (for steady conditions) in accordance with Newton’s second law for a body without any influencing forces:

ρAUθ*2πr=k

where ρ is the fluid density and A is the flow area.

This can be simplified to:

2πrUθ=Γ

where Γ is the called the circulation value, which is constant for a free vortex.

Looking at this equation it can be seen that velocity varies inversely to the radius:

Uθ=Γ/2πr

The difference in velocity between the inner and outer edge of a fluid element being carried in the flow causes it to retain its rotational orientation regardless of its position in the vortex, as shown in Figure 2.

For this reason a free vortex in also called an irrotational vortex.

Figure 2: Free vortex behaviour

Both the free and forced vortex behaviours cannot prevail in a real fluid as the velocity will approach infinity with increasing or decreasing radius for the force and free vortex behaviours respectively. This is obviously not a realistic outcome.

Dr Daniel Jarman, Technology Manager, Hydro International

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