Specific speed N, is used to characterize turbomachinery speed. Common commercial and industrial practices use dimensioned versions which are of equal utility. Specific speed is most commonly used in pump applications to define the suction specific speed [http://nptel.ac.in/courses/112104117/chapter_1/1_7.html]—a quasi non-dimensional number that categorizes pump impellers as to their type and proportions. In Imperial units it is defined as the speed in revolutions per minute at which a geometrically similar impeller would operate if it were of such a size as to deliver one gallon per minute against one foot of hydraulic head. In metric units flow may be in l/s or m/s and head in m, and care must be taken to state the units used.
Performance is defined as the ratio of the pump or turbine against a reference pump or turbine, which divides the actual performance figure to provide a unitless figure of merit. The resulting figure would more descriptively be called the "ideal-reference-device-specific performance." This resulting unitless ratio may loosely be expressed as a "speed," only because the performance of the reference ideal pump is linearly dependent on its speed, so that the ratio of [device-performance to reference-device-performance] is also the increased speed at which the reference device would need to operate, in order to produce the performance, instead of its reference speed of "1 unit."
Specific speed is an index used to predict desired pump or turbine performance. i.e. it predicts the general shape of a pumps impeller. It is this impellers "shape" that predicts its flow and head characteristics so that the designer can then select a pump or turbine most appropriate for a particular application. Once the desired specific speed is known, basic dimensions of the unit's components can be easily calculated.
Several mathematical definitions of specific speed (all of them actually ideal-device-specific) have been created for different devices and applications.