When a fluid flows steadily over an isolated cylindrical solid barrier and the Reynolds number is great than about 50, vortices are shed on the downstream side. The vortices trail behind the cylinder in two rolls, alternatively from the top or the bottom of the cylinder. This vortex trail is call the von Karman vortex street or Karman street after von Karman's 1912 mathematical description of the phenomenon.
The Karmen street has two significant influences on the principle of operation of vortex flowmeters:
- The frequency of vortex shedding is definite and is related to the Reynolds number (flow velocity, viscosity of fluid, and the diameter of the cylinder).
- The frequency of vortex shedding is the same as the vibrating frequency of the cylinder induced by the flow.
If the density and viscosity of the fluid are known and the diameter of the cylinder is given, the frequency measured at the cylinder can be used to represent the flow velocity.
Dimensional anaylsis shows that the frequency of vortex shedding fv is governed by the Strouhal number of the vortex pattern
where Dc is the diameter of the cylinder or width of the barrier and V is the flow velocity.
The Strouhal number is a dependent variable of the Reynolds number:
However, experimental investigations show that the Strouhal number is about constant across a wide range of the Reynolds number (102 ~ 107). This yields
The Strouhal number is about 0.18 for a cylinder at a Reynolds number range 300 ~ 107.
Notice that the flow velocity profile, the shape of the bluff (barrier), and the ratio of the cross section area of the bluff to the pipe will all affect the Strouhal number. One would need to incorporate a correction factor when applying the above formula.
Assume that the bluff is a rod across the diameter of the pipe, the width of the bluff is w that is equivalent to Dc in the above formula, and the diameter of the pipe is D. The flow velocity at the bluff is
For incompressible pipe flow, at any given moment, the flow rate is constant throughout the pipe, that is, . Thus, the ratio of upstream flow velocity V to the bluff region flow velocity Vbluff is
where A is the cross section area of the pipe, and Amin is the cross section area of the pipe with the obstructing bluff.
From the above equation, the flow velocity V can be written as
and the volume flow rate Q is
In industrial applications, a K factor is usually introduced to compensate for the non-uniform profile of the pipe flow. The volume flow rate then becomes
The Strouhal number S can be determined experimentally. Some common Strouhal numbers are listed below for reference purposes. The bluff is either a cylinder or a square column.
| w/D || S |
The above formulas assume a steady-state upstream flow. Disturbances on the upstream side may affect the vortex shedding frequency and result in measurement errors.