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The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest. Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal irrotational vortex , where most particles rotate about some straight axis, with speed inversely proportional to their ...
It has non-zero vorticity everywhere outside the core. Rotational vortices are also called rigid-body vortices or forced vortices. For example, if a water bucket is spun at constant angular speed w about its vertical axis, the water will eventually rotate in rigid-body fashion.
where is a constant freestream velocity far away from the origin and is the radius of the sphere within which the vorticity is non-zero. For r ≥ a {\displaystyle r\geq a} , the vorticity is zero and the solution described above in that range is nothing but the potential flow past a sphere of radius a {\displaystyle a} .
For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. In flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. [1]
The Rayleigh–Kuo criterion states that the gradient of the absolute vorticity should change sign within the domain. In the example of the shear induced eddies on the right, this means that the second derivative of the flow in the cross-flow direction, should be zero somewhere.
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field , a divergence-free vector field , or a transverse vector field ) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.}
At =, we have a potential vortex with concentrated vorticity at the axis; and this vorticity diffuses away as time passes. The only non-zero vorticity component is in the z {\displaystyle z} direction, given by
Animation of a Rankine vortex. Free-floating test particles reveal the velocity and vorticity pattern. The Rankine vortex is a simple mathematical model of a vortex in a viscous fluid. It is named after its discoverer, William John Macquorn Rankine. The vortices observed in nature are usually modelled with an irrotational (potential or free ...