2.5. TYPES OF FLUID FLOW AND REYNOLDS NUMBER

2.5A. Introduction and Types of Fluid Flow

The principles of the statics of fluids, treated in Section 2.2, are almost an exact science. On the other hand, the principles of the motions of fluids are quite complex. The basic relations describing the motions of a fluid are the equations for the overall balances of mass, energy, and momentum, which will be covered in the following sections.

These overall or macroscopic balances will be applied to a finite enclosure or control volume fixed in space. We use the term “overall” because we wish to describe these balances from outside the enclosure. The changes inside the enclosure are determined in terms of the properties of the streams entering and leaving and the exchanges of energy between the enclosure and its surroundings.

When making overall balances on mass, energy, and momentum we are not interested in the details of what occurs inside the enclosure. For example, in an overall balance, average inlet and outlet velocities are considered. However, in a differential balance, the velocity distribution inside an enclosure can be obtained by the use of Newton's law of viscosity.

In this section we first discuss the two types of fluid flow that can occur: laminar flow and turbulent flow. Also, the Reynolds number used to characterize the regimes of flow is considered. Then in Sections 2.6, 2.7, and 2.8, the overall mass balance, energy balance, and momentum balance are covered together with a number of applications. Finally, a discussion is given in Section 2.9 on the methods of making a shell balance on an element to obtain the velocity distribution in the element and the pressure drop.

2.5B. Laminar and Turbulent Flow

The type of flow occurring in a fluid in a channel is important in fluid dynamics problems. When fluids move through a closed channel of any cross section, either of two distinct types of flow can be observed, according to the conditions present. These two types of flow can commonly be seen in a flowing open stream or river. When the velocity of flow is slow, the flow patterns are smooth. However, when the velocity is quite high, an unstable pattern is observed, in which eddies or small packets of fluid particles are present, moving in all directions and at all angles to the normal line of flow.

The first type of flow, at low velocities, where the layers of fluid seem to slide by one another without eddies or swirls being present, is called laminar flow, and Newton's law of viscosity holds, as discussed in Section 2.4A. The second type of flow, at higher velocities, where eddies are present giving the fluid a fluctuating nature, is called turbulent flow.

The existence of laminar and turbulent flow is most easily visualized by the experiments of Reynolds. His experiments are shown in Fig. 2.5-1. Water was allowed to flow at steady state through a transparent pipe with the flow rate controlled by a valve at the end of the pipe. A fine, steady stream of dyed water was introduced from a fine jet as shown and its flow pattern observed. At low rates of water flow, the dye pattern was regular and formed a single line or stream similar to a thread, as shown in Fig. 2.5-1a. There was no lateral mixing of the fluid, and it flowed in streamlines down the tube. By putting in additional jets at other points in the pipe cross section, it was shown that there was no mixing in any parts of the tube and the fluid flowed in straight parallel lines. This type of flow is called laminar or viscous flow.

As the velocity was increased, it was found that at a definite velocity the thread of dye became dispersed and the pattern was very erratic, as shown in Fig. 2.5-1b. This type of flow is known as turbulent flow. The velocity at which the flow changes is known as the critical velocity.

2.5C. Reynolds Number

Studies have shown that the transition from laminar to turbulent flow in tubes is not only a function of velocity but also of density and viscosity of the fluid and the tube diameter. These variables are combined into the Reynolds number, which is dimensionless:

Equation 2.5-1

where N_Re is the Reynolds number, D the diameter in m, ρ the fluid density in kg/m³, μ the fluid viscosity in Pa · s, and μ the average velocity of the fluid in m/s (where average velocity is defined as the volumetric rate of flow divided by the cross-sectional area of the pipe). Units in the cgs system are D in cm, ρ in g/cm³, μ in g/cm · s, and ν in cm/s. In the English system D is in ft, ρ in lb_m/ft³, μ in lb_m/ft · s, and ν in ft/s.

The instability of the flow that leads to disturbed or turbulent flow is determined by the ratio of the kinetic or inertial forces to the viscous forces in the fluid stream. The inertial forces are proportional to ρν² and the viscous forces to μν/D, and the ratio ρν²/(μν/D) is the Reynolds number Dνρ/μ. Further explanation and derivation of dimensionless numbers are given in Section 3.11.

For a straight circular pipe, when the value of the Reynolds number is less than 2100, the flow is always laminar. When the value is over 4000, the flow will be turbulent, except in very special cases. In between—called the transition region—the flow can be viscous or turbulent, depending upon the apparatus details, which cannot be predicted.

EXAMPLE 2.5-1. Reynolds Number in a Pipe Water at 303 K is flowing at the rate of 10 gal/min in a pipe having an inside diameter (ID) of 2.067 in. Calculate the Reynolds number using both English units and SI units. Solution: From Appendix A.1, 7.481 gal = 1 ft3. The flow rate is calculated as From Appendix A.2, for water at 303 K (30°C), Substituting into Eq. (2.5-1). Hence, the flow is turbulent. Using SI units,