(Used with Open Channel Flow)
As discussed in another section,
dimensional analysis can be used to identify
dimensionless groups (pi terms) governing a system. Some common dimensionless
groups in fluid mechanics are introduced here.
Reynolds Number (Re): The Reynolds number perhaps is
the most common dimensionless parameter
used in fluid mechanics. It is defined as
Re = ρVL/μ
where ρ is the density, V is the velocity,
L is the characteristic length, and
μ is the viscosity. The L term is different for each flow type. For example, for a pipe, L is the diameter of the pipe. For open channel flow, the hydraulic radius, Rh (see diagram) is commonly used.
Physically, Re represents the ratio of the inertial force to the viscous force. A small Reynolds number implies that the viscous effects are important, while the inertial effects are dominant when the Reynolds number is large.
The Reynolds number is commonly used to characterize if a flow in a pipe is
laminar or turbulent. The flow is generally assumed to be laminar when Re < 2,100, and turbulent when Re > 4,000. The flow is referred to as transition flow when the Reynolds number is between 2,100 and 4,000.
Froude Number (Fr): The Froude number is an important dimensionless parameter in the study of open-channel flow, and it is given by
Fr = V / (gL)0.5
where V is the average velocity, L is the characteristic length associated
with the depth (hydraulic depth for open channel flow), and g is the gravitational acceleration. For rectangular cross sections, the hydraulic depth is the water depth. Physically, the Froude number represents
the ratio of inertial forces to gravitational forces.
As discussed in the open-channel sections, open-channel flow can
be classified according to the Froude number in the following manner:
(a) Fr < 1: subcritical (tranquil) flow
(b) Fr = 1: critical flow
(c) Fr > 1: supercritical (rapid) flow
It is also common to write Fr as V/c, where c is the wave celerity, c (speed of a wave in the fluid). This form is similar to the Mach Number in air. For subcritical flow (V < c), the waves created by any surface disturbances (e.g., throwing a stone in the water) at the downstream can travel upstream. On the other hand, for supercritical flow (V > c), all surface disturbances will be swept downstream. The wave will remain stationary for critical flow (V = c).
B-2 Spirit Bomber: High Subsonic Speed
Concorde from British Airways:
Supersonic Speed (Mach 2)
Mach Number (Ma): For high speed flows in some fluids, density is highly dependent on the pressure, and the compressibility effects become important. The Mach number is used to indicate if a flow is incompressible or compressible, and it is given by
Ma = V/c
where c is the speed of sound (343 m/s at 20oC) and V is the fluid velocity. The Mach number represents the ratio of inertia forces to compressibility forces. Flow can be characterized using the Mach number as folllows:
(a) Ma ≤ 0.3: incompressible
(b) 0.3 < Ma < 1.0: compressible subsonic flow
(c) Ma ≥ 1.0: compresible supersonic flow
The Mach number is often used to classify the top speed of a fighter or
passenger jet. For example, the B-2 bomber shown in the picture is capable of reaching
high subsonic speed. The Concorde of British Airways is a supersonic passenger
jet, which cruises at Mach 2, and it takes only approximately 4 hours from Los Angeles to Tokyo.
Compressible flow is beyond the scope of this introductory
fluid mechanics eBook, and is discussed in an advanced level
fluid mechanics course.
Weber Number (We): The dimensionless
parameter associated with surface tension effects is the Weber number, and it is defined as
We = ρV2L/σ
where σ is the surface tension. The
Weber number denotes the ratio of the inertial forces to surface tension
forces. The Weber number becomes an important parameter when dealing
with applications involve two fluid interfaces such as the flow of thin films of liquid and bubble formation.