# 4.2 Reynolds Number as an Indicator of Flow Regime

The Reynolds number, Re, is a dimensionless number used in fluid mechanics to indicate whether flow is laminar or turbulent (Reynolds, 1883). The original experiments conducted by Reynolds used straight smooth glass pipes. He injected a small dye stream into the center of the clear water stream while temperature was held constant and the flow rate through the tube could be controlled and measured (Reynolds, 1883). The experiments started with very low velocities and a stable single line of colored water passing through the tube as a distinct straight line of color, then velocity was increased in small increments until turbulence was reached and subsequently velocity was gradually reduced until laminar flow was again achieved. Numerous experiments with the same diameter tube confirmed the same velocity for the onset of turbulence from the laminar to turbulent state and a different velocity for the transition from turbulent back to laminar flow. Reynolds then conducted experiments with different diameter pipes and at different temperatures.

The Reynolds number, Re, represents the ratio of fluid inertial to viscous forces. It is a similarity law such that the onset of turbulence in any size of smooth, straight pipe will be relatively the same (typically, 2100 < Re < 2300). The critical Reynolds number, Rec, is the value of Re when flow begins to deviate from linear flow and below which flow is laminar. Re is computed and defined as shown in Equation 2.

 (2)

where:

 Re = Reynolds number (dimensionless) V = Q/A and is the mean flow velocity (LT−1) across a cross sectional area (also called Darcy velocity), which is equal to the volumetric flow, Q (L3T−1), divided by the cross-sectional area perpendicular to the direction of flow, A (L2) D = mean pore size diameter for porous media or the pipe diameter (L) ρ = density of the water (ML−3) μ = absolute or dynamic viscosity of water (ML−1T−1] ν = kinematic viscosity of water (L2T−1)

Exercise 10 invites the reader to consider why Reynolds tried experiments with different temperatures of water with each pipe. A second exercise uses the results from Exercise 6 (average velocity at different radial distances from a pumping well) to calculate the Re at each radial distance.

For porous media, the Darcy velocity (V=Q/A), not the pore velocity (which is the Darcy velocity divided by effective porosity), is used to define the Reynolds number. Porous media flow tends to remain at low velocity and laminar under many natural gradients because of the small pore diameters and the effect of surface tensional forces between the water and rock. For most porous media, the Rec ranges from 1 to 60, and is dependent on smoothness of the grains, tortuosity of the connected pore spaces, average pore diameter, temperature, as well as other properties of the aquifer and fluid. When flow becomes turbulent, some of the flow energy is lost by the movement of water in eddies, which results in specific discharge not increasing as rapidly as the head gradient increases. Thus, in turbulent conditions, flow is no longer a linear function of head gradient (Figure 36), as is the case for Equation 1, thereby limiting the applicability of Darcy’s law. Limitations to the applicability of Darcy’s law for very low permeability media is also expected, but this has not been thoroughly tested (Ingebritsen and Sanford, 1998).

Figure 36  Data from many field tests in fractured rocks reveal Darcian flow at low gradients as indicated by a linear relationship between gradient and flow (zone A of this graph). At higher gradients (zone B of this graph), an increase of gradient results in a smaller increase in flow because flow paths change direction and some flow energy is lost to the crossing flow paths as shown in Figure 34b (Lage and Antohe, 2000). Eventually, larger increases of gradient cause larger increases in flow because the higher gradient causes one of the following that increase the hydraulic conductivity of the material: short circuiting around testing seals; dilation of the fractures/pores; or new fracturing, as shown in zone C of this graph. From Quin and others (2011b). In pipes and porous media there is also a transition from laminar to turbulent flow where the flow rate begins to drop with increasing gradient, but once fully turbulent, the flow drop is a function of the velocity squared as opposed to the Darcy velocity as discussed in Section 4.4.

It is not possible to know the exact value of Rec for a specific aquifer or porous medium, but Rec can be estimated through laboratory experiments. Many researchers have measured discharge under different gradients to estimate the Rec by noting the point at which discharge becomes a nonlinear function of the gradient. The larger the pore diameter and smoother the grain surfaces, such as might occur in a well-sorted gravel, point-bar deposit, the higher the Rec. For porous media, various representative lengths are used for D in order to calculate Re, such as a representative grain size (frequently d10 or d50 is used, which is the diameter of the sieve size at which 10 or 50 percent, respectively, of the grains pass in a standardized geotechnical sieve analysis). Many textbooks report the range of Rec for porous media from 1 to 10 (for example, Bear, 1979; Freeze and Cherry, 1979). Schneebeli (1955) used glass spheres and found flow became turbulent at a range of Re from 5 to 60, which is consistent with our expectation that the transition to turbulence occurs at a higher Re for smooth surfaces with relatively uniform geometry.