Victor M. Ponce
Professor of Civil Engineering
San Diego State University
• INTRODUCTION •
The Reynolds number is a dimensionless number expressed as (Chow, 1959):
in which ρ = mass density of the fluid, V = flow velocity, L = a characteristic length, and μ = dynamic viscosity, a function of fluid type and temperature.
The conventional interpretation of the Reynolds number is that it is a ratio of inertial to viscous forces (or stresses). The inertial stress is represented as:
More properly though, Eq. 2 is an expression not of an inertial stress, which arises under unsteady flow, but rather of a steady external frictional stress. In fact, Eq. 2 is an expression for the quadratic law of (external) friction.
Drawing on Newton's law of viscosity, the viscous (or internal) stress is (Chow, 1959):
Thus, it follows that Eq. 1 is properly a ratio of external to internal stresses.
By recasting the Reynolds number as a ratio of fluid macroviscosity to microviscosity, an improved formulation can be obtained. These propositions are now substantiated.
• THE REYNOLDS NUMBER •
The Reynolds number (Eq. 1) can also be expressed in terms of kinematic viscosity (ν):
The parameter L is a characteristic length. In closed-conduit flow, L is interpreted as the pipe diameter D, such that:
In open-channel flow, L is interpreted as the hydraulic radius R. Thus, the Reynolds number is:
For a circular pipe: R = D/4. Therefore, the Reynolds number for pipe flow in terms of hydraulic radius is:
Equations 5 to 7 have been in common use for nearly a century. There is an implicit understanding of the Reynolds number as a ratio of viscosities. The numerator is effectively a macroviscosity, a function of the bulk fluid properties. The denominator is the fluid's microviscosity or internal viscosity, a function of fluid type and temperature.
• MACROVISCOSITY •
Viscosity, or diffusivity, has the units of [L2/T]. Viscosity is essentially a product of a velocity (the fluid's velocity) times a characteristic length. The question is: What should the characteristic length be in the definition of the Reynolds number?
Since the Reynolds number is effectively a ratio of macro- to microviscosity, a more appropriate question is: What is the expression for macroviscosity? This subject has been elucidated by Hayami, who developed the formula for the hydraulic diffusivity of a fluid (water) in hydraulically wide open-channel flow (Hayami, 1951):
in which q = discharge per unit of width, and S = energy slope or energy gradient (head loss per unit of length).
Since q = Vd, where d = flow depth, the hydraulic diffusivity is:
• MODIFIED REYNOLDS NUMBER •
In hydraulically wide open-channel flow, the modified Reynolds number is:
Replacing Eq. 9 in Eq. 10:
Since d ≅ R:
which should be compared with Eq. 6:
Thus, it is seen that the theoretical characteristic length to use in the Reynolds number is not R, but rather R/(2S). The distinction is significant, because while R (and d) is typically a nearly vertical dimension, R/(2S) is instead a horizontal dimension (half of the horizontal distance that it would take the steady uniform flow to drop a head equal to its hydraulic radius or flow depth).
For practical applications in open-channel flow, it is better to formulate the modified Reynolds number as:
In closed-conduit flow, a corresponding expression is:
• SUMMARY •
A modified Reynolds number is formulated for open-channel flow based on the expression of macroviscosity (hydraulic diffusivity, Eq. 8) originally due to Hayami (1951). The modified Reynolds number is:
Unlike the traditional Reynolds number, in which the characteristic length is perpendicular to the flow direction (typically almost vertical), in the modified Reynolds number the characteristic length is horizontal.
A corresponding expression for closed-conduit flow is:
It remains to be seen what effect the inclusion of energy slope into the Reynolds number will have on the transition range that separates laminar from turbulent flow.
• REFERENCES •
Hayami, S. 1951. On the propagation of flood waves. Disaster Prevention Research Institute, Bulletin No. 1, Kyoto University, Japan, December.
Chow, V. T. 1959. Open-channel hydraulics. Mc-Graw-Hill, New York.
• NOTATION •
d = flow depth;
D = pipe diameter;
L = characteristic length, length unit;
q = discharge per unit of width;
R = hydraulic radius;
Re = Reynolds number, Eq. 1 or Eq. 4;
S = energy slope or energy gradient (head loss per unit of length);
T = time unit;
V = flow velocity;
μ = dynamic viscosity;
ν = kinematic viscosity;
νh = hydraulic diffusivity, Eq. 8;
ρ = mass density of the fluid;
τi = inertial stress, Eq. 2; and
τv = viscous stress, Eq. 3.