Von Kármán swirling flow

Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921.

[1] The rotating disk acts as a fluid pump and is used as a model for centrifugal fans or compressors.

This flow is classified under the category of steady flows in which vorticity generated at a solid surface is prevented from diffusing far away by an opposing convection, the other examples being the Blasius boundary layer with suction, stagnation point flow etc.

Consider a planar disk of infinite radius rotating at a constant angular velocity

This outward radial motion of the fluid near the disk must be accompanied by an inward axial motion of the fluid towards the disk to conserve mass.

Theodore von Kármán[1] noticed that the governing equations and the boundary conditions allow a solution such that

Due to symmetry, pressure of the fluid can depend only on radial and axial coordinate

are Self-similar solution is obtained by introducing following transformation,[2] where

The self-similar equations are with boundary conditions for the fluid

are The coupled ordinary differential equations need to be solved numerically and an accurate solution is given by Cochran(1934).

[3] The inflow axial velocity at infinity obtained from the numerical integration is

, so the total outflowing volume flux across a cylindrical surface of radius

Neglecting edge effects, the torque exerted by the fluid on the disk with large (

The torque predicted by the theory is in excellent agreement with the experiment on large disks up to the Reynolds number of about

, the flow becomes turbulent at high Reynolds number.

[4] This problem was addressed by George Keith Batchelor(1951).

i.e., the fluid at infinity rotates in the same sense as the plate.

, the solution is more complex, in the sense that many-solution branches occur.

Zandbergen and Dijkstra[7][8] showed that the solution exhibits a square root singularity as

, at which point, a third-solution branch is found to emerge.

They also discovered an infinity of solution branches around the point

Bodoyni(1975)[9] calculated solutions for large negative

The flow accepts a non-axisymmetric solution with axisymmetric boundary conditions discovered by Hewitt, Duck and Foster.

and the governing equations are with boundary conditions The solution is found to exist from numerical integration for

Here the solution is not simple, because of the additional length scale imposed in the problem i.e., the distance

In addition, the uniqueness and existence of a steady solution are also depend on the corresponding Reynolds number

Thus, instead of the scalings used before, it is convenient to introduce following transformation, so that the governing equations become with six boundary conditions and the pressure is given by Here boundary conditions are six because pressure is not known either at the top or bottom wall;

, Batchelor argued that the fluid in the core would rotate at a constant velocity, flanked by two boundary layers at each disk for

There is also an exact solution if the two disks are rotating about different axes but for

Von Kármán swirling flow finds its applications in wide range of fields, which includes rotating machines, filtering systems, computer storage devices, heat transfer and mass transfer applications, combustion-related problems,[15] planetary formations, geophysical applications etc.