What is the difference between Single jet meters & Multi jet meters?

Single jet meters make use of only one port to create a jet of water, making the turbine rotate, whereas Multi‐jet meters use multiple ports surrounding an internal chamber to create a jet of water against the turbine. Both, single jet and multi‐jet meters are widely used. Accuracy and flow range requirements are same for single jet and multi jet water meters.

Single jet meters are perhaps the first of the modern flowmeters for water.
The flow path is tangential to the rotor and there is no flow through the rotor. It would seem as if this is based on the vorticity principle in which the energy is transferred from the flow stream to the rotor through fluid boundary layer friction.

This means that the meter is basically suited for turbulent flow where the flow velocity adjacent to the rotor is high. In laminar flow the flow velocity profile does not allow for any significant energy transfer into the vortex the rotor creates.

These early meters and indeed through most of the18th century had a number of problems. In a paper given in the 1870s it was pointed out that there is a delay between flow starting and the meter responding, perhaps because the initial energy transfer creates vorticies within only one or two compartments between pairs of blades before enough energy is transferred to initiate rotation?

There was also a problem of a delay between flow stopping and the energy on the vortex decaying… a feature of vortexes that they are stable once created. The only comment made at this time was that “Dr Siemens assures us these delays cancel out.” A polite way off saying he produced no evidence to substantiate this claim?

In the multi-jet meter the rotor spins inside a chamber with flow introduced via angled ports in the chamber walls designed to ensure the the responsiveness of the rotor to flow.
Over time a number of innovations helped improve both designs including fan chamber breaks in the multi-jet to break up the votex that remains when flow stops and the fan keeps rotating.

There are also exhaust baffles or brakes intended to help prevent various other effects. If we imagine the rotor spinning at high enough speed, there comes a tendency to create cavitation in the centre? Not sure on that one. As with all such designs we need to make an effort to understand exactly what is happening to the flow.