Chapter 14

The Reynolds number

14.1 The Reynolds number from dimensional analysis

Carrying out several experiments with different values of the pipe radius R, the water viscosity ν and the pressure gradient, it is possible to verify that the transition to turbulence in the Reynolds experiment occurs for the same value of the dimensionless parameter UR Re = , ν where U is the mean velocity of the flow. This quantity is called the Reynolds number. This result is in accord with the general rule that any physical law must depend on dimensionless parameters, because it cannot depend on the chosen units of measurement. And the Reynolds number is the only number that we can construct on the basis of the dimensional quantities (R, ν and U) that define the problem. More precisely, we never have turbulence for Reynolds numbers less than about 1 000, even in the presence of an incoming disturbed flow. The intermit- tent regime exists from < = 1000 to < = 2000. The laminar regime can be maintained up to a Reynolds number equal to 50000, if particular care is taken in the construction of the experimental apparatus.

103 104 Franco Mattioli (University of Bologna)

In the Reynolds original experiment the transition to turbulence occurred for < = 1150. Later, Reynolds himself managed to obtain a laminar flow up to < = 6500.

Problem 14.1 Evaluate the Reynolds number relative to a pipe of a section of 2 squared centimeters (corresponding to a radius of 8 mm) in which the mean velocity of the water flow is 25 cm s−1 (in such a way to fill a liter bottle of water in 20 s). Solution. The Reynolds number is

UR 0.25 × 8 × 10−3 Re = = =2 × 103, ν 10−6 The flow inside the pipe will be certainly turbulent, although for lower velocities, of the order of those present in a typical kitchen sink, it might also be laminar. Comment. This computation refers only to the flow inside the pipe, and not to the exiting flow, which is subject to a series of other complicated physical processes.

The definition of Reynolds number introduced above can be extended to any other flow which has a structure similar to the flow in a pipe, that is, any flow in which the mean velocity in one direction varies strongly in the other two coordinates. This situation occurs frequently in geophysics. It is easy to verify that in most cases such meteo-oceanographic flows are turbulent. The meridional branch of the Gulf Stream has a typical magnitude of 1 m s−1 and its core is placed at a distance of the order of 100 km, so that the Reynolds number is

UR 100 × 103 Re = = = 1011, ν 10−6 where we have assumed ν = 10−6 m2 s−1. As can be seen, the motion must be turbulent. The typical magnitude of the wind velocity at a height of 1 km is10ms−1, so that

UR 10 × 103 Re = = =7 × 108, ν 1.5 × 10−5 where we have assumed ν =1.5 × 10−5 m2 s−1. In this case, too, the turbulent regime is assured. Indeed, these examples refer to problems which are rather different from a flow in a pipe. The domain in which the turbulent motion develops is only a finite part of the whole domain occupied by the fluid and additional physical factors are present, such as the effects of the earth’s rotation and other thermodynamic processes. Nevertheless, the transition to turbulence is still strongly influenced by the Reynolds number, i.e., by a simple mechanical parameter of the motion. Elements of Fluid Dynamics (www.fluiddynamics.it) 105

14.2 The Reynolds number from scale analysis

We can understand why a low Reynolds number implies stability and a high one instability by means of a scale analysis, that is, a comparison between the orders of magnitude of certain terms of the momentum equation (10.6), whose component in the x direction reads 2 2 2 ∂u ∂u ∂u ∂u −1 ∂p ∂ u ∂ u ∂ u + u + v + = + ν 2 + 2 + 2 . (14.1) ∂t ∂x ∂y ∂z ! ρ ∂x ∂x ∂y ∂z ! Since the largest turbulent fluctuations are isotropic, this equation is representa- tive of the other two equations in the y and z directions and the terms between parenthesis are of the same order of magnitude. It is experimentally seen that the magnitude of the largest fluctuations is of the order of the radius R of the pipe and that the magnitude of the various components of the velocity fluctuations is of the same order of the mean velocity U. Thus, the order of magnitude of the nonlinear terms becomes 2 ∂u U u ∼ , ∂x R while that of the viscous terms is ∂2u U ν ∼ ν . ∂x2 R2

The Reynolds number turns out to be the ratio of these two orders of magnitude 2 2 U R UR = = Re. R νU ν

It is worth noticing that here the Reynolds number, although defined as the ratio of two quantities referring to the turbulent fluctuations, is indeed written in terms of quantities relative to the macroscopic properties of the flow. The presence of turbulence requires an high Reynolds number, so that the advective terms must be greater than the dissipative terms associated to the turbulent fluctuations. Since the viscosity gives rise to a process of attenuation of the velocity fluctuations, the cause of the turbulence must lie in the nonlinear terms of the momentum equation. If this last process is not sufficiently strong, then turbulence is inhibited. But if the viscous damping is weak, then we have the onset of turbulence. 106 Franco Mattioli (University of Bologna)

14.3 The Reynolds number from the properties of the mean flow

The profile of the mean velocity, shown in (Fig. 13.1) is characterized by a sharp variation near the walls, where the instantaneous velocity must vanish. In the thin boundary layer near the walls the turbulent fluctuations are inhibited by the presence of the wall itself. Thus, in its vicinity, their magnitude is very small and a thin layer of an essentially laminar flow must exist. We can derive the order of magnitude of the thickness of this viscous boundary layer by means of the dimensional analysis. This thickness δv must depend on the kinematic viscosity ν of the fluid and on the mean velocity U of the flow, so that ν δv = . U Then, we can write R < = . δv Thus, the Reynolds number represents the ratio between the radius of the pipe and the order of magnitude of the viscous boundary layer. As the Reynolds number increases, this laminar boundary layer becomes increasingly thin.