Womersley Number - Derivation

Derivation

The Womersley number, usually denoted, is defined by the relation

where is an appropriate length scale (for example the radius of a pipe), is the angular frequency of the oscillations, and, are the kinematic viscosity, density, and dynamic viscosity of the fluid, respectively. The Womersley number is normally written in the powerless form

It can also be written in terms of the dimensionless Reynolds number (Re) and Strouhal number (Sr):

The Womersley number arises in the solution of the linearized Navier Stokes equations for oscillatory flow (presumed to be laminar and incompressible) in a tube. It expresses the ratio of the transient or oscillatory inertia force to the shear force. When is small (1 or less), it means the frequency of pulsations is sufficiently low that a parabolic velocity profile has time to develop during each cycle, and the flow will be very nearly in phase with the pressure gradient, and will be given to a good approximation by Poiseuille's law, using the instantaneous pressure gradient. When is large (10 or more), it means the frequency of pulsations is sufficiently large that the velocity profile is relatively flat or plug-like, and the mean flow lags the pressure gradient by about 90 degrees. Along with the Reynolds number, the Womersley number governs dynamic similarity.

The boundary layer thickness that is associated with the transient acceleration is related to the Womersley number. It is equal to inverse of the Womersley number. The Womersley number is also equal to the square root of the Stokes number.

where L is a characteristic length.