Due to the relative age of the technology, it may be beneficial to discuss the operation of the multipath ultrasonic meter to illustrate the effects of flow profile distortion and swirl. There are various types of flow measurements utilizing high frequency sound. The custody transfer measurement devices available today utilize the time of travel concept. The difference in time of flight with the flow is compared to the time of flight against the flow. This difference is used to infer average flow velocity on the sound path. Fig.(5) showing the Ultrasonic Meter sound path no flow which illustrates this concept.
The resulting flow equation for the mean velocity experienced by the sound path is given by,
    ----(5)<br />
The case of no flow gives the actual path of the sound when there is zero flow (by equating eq.(5) to zero). In case of theoretical flow profile, say a uniform velocity flow profile where the no-slip condition on the pipe walls is not applied, Fig.(6) shows Ultrasonic Meter sound path - uniform velocity profile which illustrates the resultant sound path.
A theoretical derivation of the Mean velocity equation for this sound path becomes much more complicated. In case of a perfect fully developed real velocity profile of Ultrasonic meter which is shown in Fig.(7) indicating a possible sound path as a result of an installation in a real flow. 
Here a mathematical derivation for this Ultrasonic meter is also becomes very complicated. Developing a robust flow algorithm to calculate the mean flow velocity for the sound path can be quite complicated. Now add to this; sound path reflection from the pipe wall, multipaths to add degrees of freedom, swirl and departure from axisymmetric fully developed flow profile and the problem of integrating the actual velocity flow profile to yield volume flow rate can be an accomplishment. Hence the real performance of ultrasonic meters downstream of perturbations, and the need for calibrations is required.