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Reynolds number

A vortex street around a cylinder. This occurs around cylinders, for any fluid, cylinder size and fluid speed, provided that there is a Reynolds number of between ~40 and 103.[1]

In

  • The Reynolds Number at Sixty Symbols

External links

  • Zagarola, M. V. and Smits, A. J., "Experiments in High Reynolds Number Turbulent Pipe Flow." AIAA paper #96-0654, 34th AIAA Aerospace Sciences Meeting, Reno, Nevada, January 15–18, 1996.
  • Jermy M., "Fluid Mechanics A Course Reader," Mechanical Engineering Dept., University of Canterbury, 2005, pp. d5.10.
  • Hughes, Roger "Civil Engineering Hydraulics," Civil and Environmental Dept., University of Melbourne 1997, pp. 107–152
  • Fouz, Infaz "Fluid Mechanics," Mechanical Engineering Dept., University of Oxford, 2001, p. 96
  • E. M. Purcell. "Life at Low Reynolds Number", American Journal of Physics vol 45, pp. 3–11 (1977)[1]
  • Truskey, G. A., Yuan, F, Katz, D. F. (2004). Transport Phenomena in Biological Systems Prentice Hall, pp. 7. ISBN 0-13-042204-5. ISBN 978-0-13-042204-0.

Further reading

  1. ^ Tansley, Claire E.; Marshall, David P. (2001). "Flow past a Cylinder on a Plane, with Application to Gulf Stream Separation and the Antarctic Circumpolar Current". Journal of Physical Oceanography 31 (11): 3274–3283.  
  2. ^  
  3. ^  
  4. ^ a b Rott, N. (1990). "Note on the history of the Reynolds number".  
  5. ^ Falkovich, G. (2011). Fluid Mechanics. Cambridge University Press. 
  6. ^ Reynolds Number
  7. ^ Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. pp. 211–215. 
  8. ^ "Reynolds Number". Engineeringtoolbox.com. 
  9. ^ Holman, J. P. Heat Transfer. McGraw Hill. 
  10. ^ Fox, R. W.; McDonald, A. T.; Pritchard, Phillip J. (2004). Introduction to Fluid Mechanics (6th ed.). Hoboken: John Wiley and Sons. p. 348.  
  11. ^ Streeter, V. L. (1962). Fluid Mechanics (3rd ed.). McGraw-Hill. 
  12. ^ P. B. S. Lissaman (1983). "Low-Reynolds-Number Airfoils,". Ann. Rev. Fluid Mech. (15): 223–39.  
  13. ^ ISO. "International Standard Atmosphere". eng.cam.ac.uk. 
  14. ^ Uwe Ehrenstein; Christophe Eloy. "Skin friction on a moving wall and its implications for swimming animals" (PDF). Retrieved 11 February 2014. 
  15. ^ a b Rhodes, M. (1989). Introduction to Particle Technology. Wiley.  
  16. ^ Dusenbery, David B. (2009). Living at Micro Scale. Cambridge, Mass: Harvard University Press. p. 49.  
  17. ^ a b Dwivedi, P. N. (1977). "Particle-fluid mass transfer in fixed and fluidized beds". Industrial & Engineering Chemistry Process Design and Development 16 (2): 157–165.  
  18. ^ Sinnott, R. K. Coulson & Richardson's Chemical Engineering, Volume 6: Chemical Engineering Design (4th ed.). Butterworth-Heinemann. p. 73.  
  19. ^ De Witt, D. P. (1990). Fundamentals of Heat and Mass Transfer. New York: Wiley. 
  20. ^ Holman, J. P. (2002). Heat Transfer. McGraw-Hill. p. 207. 
  21. ^ Shih, Merle C. Potter, Michigan State University, David C. Wiggert, Michigan State University, Bassem Ramadan, Kettering University ; with Tom I-P. (2012). Mechanics of fluids (Fourth edition. ed.). p. 105.  

References

  1. ^ The definition of the Reynolds number is not to be confused with the Reynolds equation or lubrication equation.
  2. ^ Full development of the flow occurs as the flow enters the pipe, the boundary layer thickens and then stabilizes after several diameters distance into the pipe.

Notes

See also

This is why mathematically all newtonian, incompressible flows with the same Reynolds number are comparable. Notice also, in the above equation, as \mathrm{Re} \to \infty the viscous terms vanish. Thus, high Reynolds number flows are approximately inviscid in the free-stream.

\frac{D \mathbf{v}}{D t} = -\nabla p + \frac{1}{\mathrm{Re}} \nabla^2 \mathbf{v} + \mathbf{f}.

Finally, dropping the primes for ease of reading:

where the term \frac{\mu}{\rho L V} = \frac{1}{\mathrm{Re}}.

\frac{D \mathbf{v'}}{D t'} = -\nabla' p' + \frac{\mu}{\rho L V} \nabla'^2 \mathbf{v'} + \mathbf{f'}

we can rewrite the Navier-Stokes equation without dimensions:

\mathbf{v'} = \frac{\mathbf{v}}{V},\ p' = p\frac{1}{\rho V^2}, \ \mathbf{f'} = \mathbf{f}\frac{L}{V^2}, \ \frac{\partial}{\partial t'} = \frac{L}{V} \frac{\partial}{\partial t}, \ \nabla' = L \nabla

If we now set:

  • V \, is the mean velocity, v \, or \mathbf{\bar v} , relative to the fluid (m/s).
  • L \, is the characteristic length (m).
  • \rho \, is the fluid density (kg/m³).

where:

L \over \rho V^2

Each term in the above equation has the units of a "body force" (force per unit volume) with the same dimensions of a density times an acceleration. Each term is thus dependent on the exact measurements of a flow. When one renders the equation nondimensional, that is when we multiply it by a factor with inverse units of the base equation, we obtain a form which does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the following factor:

\rho \frac{D \mathbf{v}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{f}.

The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier-Stokes equations for a newtonian fluid expressed in the lagrangian derivative:

Derivation

Using the definition of the Reynolds number we can see that a large diameter with rapid flow, where the density of the blood is high, tends towards turbulence. Rapid changes in vessel diameter may lead to turbulent flow, for instance when a narrower vessel widens to a larger one. Furthermore, a bulge of atheroma may be the cause of turbulent flow, where audible turbulence may be detected with a stethoscope.

Poiseuille's law on blood circulation in the body is dependent on laminar flow. In turbulent flow the flow rate is proportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressure gradient in laminar flow.

In physiology

What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important at large scales of the flow. With a strong predominance of inertial forces over viscous forces, the largest scales of fluid motion are undamped—there is not enough viscosity to dissipate their motions. The kinetic energy must "cascade" from these large scales to progressively smaller scales until a level is reached for which the scale is small enough for viscosity to become important (that is, viscous forces become of the order of inertial ones). It is at these small scales where the dissipation of energy by viscous action finally takes place. The Reynolds number indicates at what scale this viscous dissipation occurs.

In a turbulent flow, there is a range of scales of the time-varying fluid motion. The size of the largest scales of fluid motion (sometimes called eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number. As the Reynolds number increases, smaller and smaller scales of the flow are visible. In a smoke stack, the smoke may appear to have many very small velocity perturbations or eddies, in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in the flow. The higher the Reynolds number, the greater the range of scales. The largest eddies will always be the same size; the smallest eddies are determined by the Reynolds number.

Smallest scales of turbulent motion

quantities marked with 'm' concern the flow around the model and the others the actual flow. This allows engineers to perform experiments with reduced models in water channels or wind tunnels, and correlate the data to the actual flows, saving on costs during experimentation and on lab time. Note that true dynamic similitude may require matching other dimensionless numbers as well, such as the Mach number used in compressible flows, or the Froude number that governs open-channel flows. Some flows involve more dimensionless parameters than can be practically satisfied with the available apparatus and fluids, so one is forced to decide which parameters are most important. For experimental flow modeling to be useful, it requires a fair amount of experience and judgement of the engineer.

\mathrm{Eu}_m = \mathrm{Eu} \; \quad\quad \mbox{i.e.} \quad {p_m \over \rho_m {v_m}^{2}} = {p\over \rho v^{2}} \; ,
\mathrm{Re}_m = \mathrm{Re} \;

In order for two flows to be similar they must have the same geometry, and have equal Reynolds numbers and Euler numbers. When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds:

Similarity of flows

Pressure drops seen for fully developed flow of fluids through pipes can be predicted using the Moody diagram which plots the Darcy–Weisbach friction factor f against Reynolds number {\mathrm{Re}} and relative roughness \epsilon / D. The diagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases. The nature of pipe flow is strongly dependent on whether the flow is laminar or turbulent.

The Moody diagram, which describes the Darcy–Weisbach friction factor f as a function of the Reynolds number and relative pipe roughness.

Pipe friction

These transition Reynolds numbers are also called critical Reynolds numbers, and were studied by Osborne Reynolds around 1895.[4] The critical Reynolds number is different for every geometry.[21]

For flow in a pipe of diameter D, experimental observations show that for "fully developed" flow,[n 2] laminar flow occurs when \mathrm{Re}_D < 2100 and turbulent flow occurs when \mathrm{Re}_D > 4000.[20] In the interval between 2300 and 4000, laminar and turbulent flows are possible and are called "transition" flows, depending on other factors, such as pipe roughness and flow uniformity. This result is generalized to non-circular channels using the hydraulic diameter, allowing a transition Reynolds number to be calculated for other shapes of channel.

In boundary layer flow over a flat plate, experiments confirm that, after a certain length of flow, a laminar boundary layer will become unstable and turbulent. This instability occurs across different scales and with different fluids, usually when \mathrm{Re}_x \approx 5 \times 10^5,[19] where x is the distance from the leading edge of the flat plate, and the flow velocity is the freestream velocity of the fluid outside the boundary layer.

Transition and turbulent flow

The system is fully turbulent for values of Re above 10 000.[18]

\mathrm{Re} = .

In a cylindrical vessel stirred by a central rotating paddle, turbine or propeller, the characteristic dimension is the diameter of the agitator D. The velocity is N D where N is the rotational speed. Then the Reynolds number is:

Stirred vessel

Laminar conditions apply up to Re = 10, fully turbulent from 2000.[15]

The choice of equation depends on the system involved: the first is successful in correlating the data for various types of packed and fludized beds, the second Reynolds number suits for the liquid-phase data, while the third was found successful in correlating the fludized bed data, being first introduced for liquid fluidized bed system.[17]

\mathrm{Re} = .

or

\mathrm{Re} = .

or

\mathrm{Re} = .

For fluid flow through a bed of approximately spherical particles of diameter D in contact, if the "voidage" is ε and the "superficial velocity" is vs, the Reynolds number can be defined as:[17]

Packed bed

The particle Reynolds number is important in determining the fall velocity of a particle. When the particle Reynolds number indicates laminar flow, Stokes' law can be used to calculate its fall velocity. When the particle Reynolds number indicates turbulent flow, a turbulent drag law must be constructed to model the appropriate settling velocity.

Fall velocity

The equation for an oblong object is identical to that of a sphere, with the object being approximated as an ellipsoid and the axis of length being chosen as the characteristic length scale. Such considerations are important in natural streams, for example, where there are few perfectly spherical grains. For grains in which measurement of each axis is impractical, sieve diameters are used instead as the characteristic particle length-scale. Both approximations alter the values of the critical Reynolds number.

Oblong object in a fluid

Under the condition of low Re, the relationship between force and speed of motion is given by Stokes' law.[16]

For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid some distance away from the sphere, such that the motion of the sphere does not disturb that reference parcel of fluid. The density and viscosity are those belonging to the fluid.[15] Note that purely laminar flow only exists up to Re = 10 under this definition.

Sphere in a fluid

It is, however, a problem for mixing of polymers, because turbulence is needed to distribute fine filler (for example) through the material. Inventions such as the "cavity transfer mixer" have been developed to produce multiple folds into a moving melt so as to improve mixing efficiency. The device can be fitted onto extruders to aid mixing.

The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who exude viscous solutions from their skin to aid flow over their bodies while swimming. It has been used in yacht racing by owners who want to gain a speed advantage by pumping a polymer solution such as low molecular weight polyoxyethylene in water, over the wetted surface of the hull.

Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The Reynolds number is very small and Stokes' Law can be used to measure the viscosity of the fluid. Spheres are allowed to fall through the fluid and they reach the terminal velocity quickly, from which the viscosity can be determined.

Creeping flow past a sphere: streamlines, drag force Fd and force by gravity Fg.

In viscous fluids

The Reynolds number for an object in a fluid, called the particle Reynolds number and often denoted Rep, is important when considering the nature of the surrounding flow, whether or not vortex shedding will occur, and its fall velocity.

Qualitative behaviors of fluid flow over a cylinder depends to a large extent on Reynolds number; similar flow patterns often appear when the shape and Reynolds number is matched, although other parameters like surface roughness have a big effect

Object in a fluid

Reynolds numbers are used in airfoil design to (among other things) manage "Scale Effect" when computing/comparing characteristics (a tiny wing, scaled to be huge, will perform differently).[12] Fluid dynamicists define the chord Reynolds number, R, like this: R = Vc / ν where V is the flight speed, c is the chord, and ν is the kinematic viscosity of the fluid in which the airfoil operates, which is 1.460x10−5 m2/s for the atmosphere at sea level.[13] In some special studies a characteristic length other than chord may be used; rare is the "span Reynolds number" which is not to be confused with span-wise stations on a wing where chord is still used.[14]

Flow around airfoils

For flow of liquid with a free surface, the hydraulic radius must be determined. This is the cross-sectional area of the channel divided by the wetted perimeter. For a semi-circular channel, it is half the radius. For a rectangular channel, the hydraulic radius is the cross-sectional area divided by the wetted perimeter. Some texts then use a characteristic dimension that is four times the hydraulic radius, chosen because it gives the same value of Re for the onset of turbulence as in pipe flow,[11] while others use the hydraulic radius as the characteristic length-scale with consequently different values of Re for transition and turbulent flow.

Flow in an open channel

For a fluid moving between two plane parallel surfaces—where the width is much greater than the space between the plates—then the characteristic dimension is twice the distance between the plates.[10]

Flow in a wide duct

For calculations involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy.

D_o is the inside diameter of the outside pipe, and
D_i is the outside diameter of the inside pipe.

where

\mathrm{Re} = = = = D_o - D_i

For flow in a pipe or tube, the Reynolds number is generally defined as:[8]

Flow in pipe

Note that multiplying the Reynolds number by \rho L \mathrm{v} \over \rho L \mathrm{v} yields \rho \mathrm{v}^2L^2 \over \mu \mathrm{v}L , which is the ratio of the inertial forces to the viscous forces.[7] It could also be considered the ratio of the total momentum transfer to the molecular momentum transfer.

  • {\mathbf v} is the mean velocity of the object relative to the fluid (SI units: m/s)
  • {L} is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter when dealing with river systems) (m)
  • {\mu} is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s))
  • {\bold \nu} is the kinematic viscosity ({\bold \nu} = \mu /{\rho}) (m²/s)
  • {\rho}\, is the density of the fluid (kg/m³).

where:

\mathrm{Re} = \dfrac{ \mbox{inertial forces} }{ \mbox{viscous forces} } = = [6]

The Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface.[n 1] These definitions generally include the fluid properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension. This dimension is a matter of convention – for example radius and diameter are equally valid to describe spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today. Other shapes such as rectangular pipes or non-spherical objects have an equivalent diameter defined. For fluids of variable density such as compressible gases or fluids of variable viscosity such as non-Newtonian fluids, special rules apply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels. The Reynolds number is defined below for each case.

Definition

Contents

  • Definition 1
    • Flow in pipe 1.1
    • Flow in a wide duct 1.2
    • Flow in an open channel 1.3
    • Flow around airfoils 1.4
    • Object in a fluid 1.5
      • In viscous fluids 1.5.1
      • Sphere in a fluid 1.5.2
      • Oblong object in a fluid 1.5.3
      • Fall velocity 1.5.4
    • Packed bed 1.6
    • Stirred vessel 1.7
  • Transition and turbulent flow 2
  • Pipe friction 3
  • Similarity of flows 4
  • Smallest scales of turbulent motion 5
  • In physiology 6
  • Derivation 7
  • See also 8
  • Notes 9
  • References 10
  • Further reading 11
  • External links 12

In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic, and very small changes to shape and surface roughness can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used.

  • laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
  • turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.[5] Reynolds numbers frequently arise when performing scaling of fluid dynamics problems, and as such can be used to determine dynamic similitude between two different cases of fluid flow. They are also used to characterize different flow regimes within a similar fluid, such as laminar or turbulent flow:

[4][3] (1842–1912), who popularized its use in 1883.Osborne Reynolds but the Reynolds number is named after [2]

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