Reynolds Number

4 Key excerpts on "Reynolds Number"

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  Life in Moving Fluids

  The Physical Biology of Flow - Revised and Expanded Second Edition

  • Steven Vogel(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  At the very least we ought to ask whether some of the variables have the same effects as others, whether each need be regarded as behaving in a com- pletely unprecedented fashion. This latter possibility turns out to be both real and useful. Its pursuit leads to the peculiarly powerful Reynolds Number, the centerpiece of biolog- ical (and even nonbiological) fluid mechanics. The utility of the Reynolds Number extends far beyond mere problems of drag; it's the nearest thing we have to a completely general guide to what's likely to happen when solid and fluid move with respect to each other. For a biologist, dealing with systems that span an enormous size range, the Reynolds Number is the central scaling parameter that makes order of a diverse set of physical phenomena. It plays a role comparable to that of the surface-to-volume ratio in physiology. This almost magical variable can be most easily introduced in just the way it originated, in the empirical investigation done by Osborne Reynolds (1883) already mentioned in Chapter 3 in connection with laminar and turbulent flow. As you may recall, Reynolds introduced a dye stream into a pipe of flowing liquid. Sometimes the resulting straight streak indicated laminar flow, and sometimes dispersal of the streak signaled that the flow was turbulent. The transition was fairly sudden, both in location in the pipe and as the characteristics of the flow were altered. He found that the flow could be persuaded to shift from laminar to turbulent in several ways: by 84 DRAG, SCALE, Reynolds Number increasing speed; by increasing the diameter of the pipe; by increasing the density of the liquid; or by decreasing the liquid's viscosity. Each change was as effective, quantitatively, as any other; and they worked in combination as well as individually. The rule that emerged was that when a certain combination of these variables exceeded 2000, the flow became turbulent.

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  Introductory Fluid Mechanics

  • Joseph Katz(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  The third nondimensional number 216 Dimensional Analysis and High-Reynolds-Number Flows is the Euler number, which represents the ratio between the pressure and the inertia forces: Eu = p 0 ρ V 2 . (6.11) A frequently used quantity that is related to the Euler number is the pressure coef-ficient C p , which measures the nondimensional pressure difference, relative to a reference pressure p 0 : C p ≡ p − p 0 1 2 ρ V 2 (6.12) The last nondimensional group in Eq. (6.7) represents the ratio between the inertial and the viscous forces and is called the Reynolds Number, which was introduced in the previous chapter: Re = ρ VL μ = VL ν . (6.13) Here ν is the kinematic viscosity, which is often used for sake of brevity: ν = μ/ρ. (6.14) For the flow of gases, from the kinetic theory point of view (see [4, p. 257] in Chap-ter 2), the viscosity can be connected to the average velocity of the molecules c and to the mean distance λ that they travel between collisions (mean free path) by μ ≈ ρ c λ 2 . Substituting this into Eq. (6.13) yields Re ≈ 2( V / c )( L /λ ) . This formulation shows that the Reynolds Number represents the scaling of the velocity times length compared with the molecular scale. Note that c is larger than the speed of sound (see Fig. 1.16). The conditions for neglecting the viscous terms when Re 1 is discussed in more detail in the next section. For simplicity, at the beginning of this analysis an incompressible fluid was assumed. However, if com-pressibility is to be considered, an additional nondimensional number appears that is called the Mach number, and it is the ratio of the velocity to the speed of sound a [see Eq. (1.33) for evaluating the speed of sound for an ideal gas]: M = V / a . (6.15) Usually when the characteristic velocity V is much less than the speed of sound, then the flow can be considered as incompressible (e.g., a car traveling at 150 km/h).

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  Unit Processes in Pharmacy

  Pharmaceutical Monographs

  • David Ganderton, J.B. Stenlake(Authors)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  The core of fluid is turbulent. In a thin layer at the wall a fraction of a millimetre thick, laminar conditions persist. This is called the laminar sub-layer and it is separated from the turbulent core by a buffer layer in which transition from turbulent to laminar flow occurs. This description of the turbulent boundary layer applies generally to the flow of fluids over surfaces. The properties of this layer are central in many aspects of the flow of fluids. In addition, these properties determine the rate at which heat or mass are transferred to or from the boundary. These subjects form Chapters 2 and 3 of the monograph. 27 UNIT PROCESSES IN PHARMACY The Significance of Reynolds Number, Re In Reynold's experiment, described on page 18, progressive increase in speed caused a change from laminar to turbulent flow. This change would also have occurred if the diameter of the tube was increased while maintaining the velocity, or if the fluid was replaced by one of higher density. On the other hand, an increase in viscosity could promote a change in the opposite direction. Obviously, all these factors are simultaneously determining the nature of flow. These factors, which alone determine the character of flow, combine to give some value of Re. This indicates that the forces acting on some fluid element have a particular pattern. If some other geometrically similar system has the same Re, the fluid will be subject to the same force pattern. More specifically, Reynolds Number describes the ratio of the inertia and viscous or frictional forces. The higher the Reynolds Number, the greater will be the relative contribution of inertial effects. At very low Re, viscous effects predominate and the con-tribution of inertial forces can be ignored. A clear example of the changing contributions of viscous and inertia or momentum effects and the resulting changes in the flow pattern is given in Fig.

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  Automotive Aerodynamics

  • Joseph Katz(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Fig. 4.1 .

  Figure 4.1

  Some typical fluid flows and their Reynolds Number. High speed sometimes means high Mach number and therefore the relevant Mach number range is also presented

  Based on the large variety of cases shown in Fig. 4.1 , we can conclude that for high Reynolds Number flows, the viscous terms become small compared to the other terms of order O(1) in Eq. (4.16) . But before neglecting these terms, a closer look at the high Reynolds Number flow condition is needed. As an example, consider the flow near a streamlined body, as shown in Fig. 4.2 . In general, based on the assumption of a high Reynolds Number, the viscous terms in the momentum equations can be neglected in the outer flow regions (outside the immediate vicinity of a solid surface). Therefore, in this outer flow region, the solution can be approximated by solving the incompressible continuity and the Euler equations:

  (4.18)

  (4.19)

  Figure 4.2

  Two major flow regimes in an attached, high-Reynolds Number flow. (1) The outer mostly inviscid and (2) the thin inner region, shown by the dashed lines, dominated by viscous effects

  Equations 4.18 –4.19

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