Reynolds number

See also: Reynolds (homonymy)

The Reynolds number characterizes a flow, and in particular the nature of its mode (laminar, transitory, turbulent etc…). It represents the relationship between inertias and viscous forces or the report/ratio (qualitative) of the transfer by convection by the transfer by diffusion of the momentum. It is most important Nombre without dimension in Dynamique of the fluids. It was highlighted in 1883 by Osborne Reynolds. It is stated generally in the following way:

Re = {\ rho V L \ over \ eta}

or

Re = {V L \ over \ naked} \; .

with (units S.I are between hooks):

  • V - speed of the fluid,
  • L - dimension characteristic of the phenomenon:
. diameter for a control (of circular section generally), hydraulic diameter.
. dimension considered to be most relevant for a control or an obstacle of an unspecified form,
. X-coordinate since the leading edge for a plane plate or a profile of wing.

In Magnétohydrodynamique it is also possible to define a Reynolds number : the Reynolds number magnetic.

Interpretation of the Reynolds number

The Reynolds number can be written in the following way:

Re = , It is interpreted then like the relationship between viscous force and inertias.

Three principal modes are distinguished.

  • With the low values of the Reynolds (lower than 2000), the forces of viscosity are dominating, convective acceleration being neglected. One speaks about flow of Stokes. The flow is laminar (nearby elements of fluid remain close). Moreover, as inertia is negligible, the flow of the fluid is reversible. That gives place to surprising behaviors: if the external forces are suddenly stopped, the fluid stops immediately. Who more is, if the external forces are reversed, the fluid sets out again in opposite direction: in a famous experiment of G.I.Taylor, a drop of ink, intialement mixed in a viscous fluid, is reconstituted when the movement was reversed.
  • With the intermediate values of the Reynolds (between 2000 and approximately 3000), the inertias are dominating, but the flow remains laminar. However, it is not reversible any more: if the external forces are stopped, the fluid continues partially on its impetus.
  • With the strong values of the Reynolds (beyond from approximately 3000, even higher), the inertias are so important that the flow becomes turbulent. Between the modes laminar and turbulent, one speaks about transitory mode.

Examples

  • In a control, the flow is laminar when the Reynolds number is lower than a value criticizes for which occurs a rather brutal transition towards the turbulent one. 2300 is the value generally retained for this transition but, under neat conditions (particularly smooth wall, stability speed), the transition can occur for a higher value. It is often considered that the transition can occur between 2000 and 3000.

  • On a cylinder with circular section placed in a flow, one obtains a properly laminar flow which is adjusted perfectly with the obstacle until a Reynolds number of about 1; a turbulent wake appears with the downstream around 105. Between the two, the transition is done through the various shapes of swirling wakes.

  • With a plane plate located in the bed of the flow, characteristic dimension is not any more the thickness of this one but the distance from a point at the edge of attack. Indeed a Boundary layer, in which viscosity or turbulence intervenes, develops starting from the leading edge. If this one presents a blunted edge, the boundary layer is turbulent as of the beginning. In the case of a frayed edge, the boundary layer is laminar over a certain length, more turbulent then. This laminar character is maintained until the distance which corresponds to the Reynolds criticizes about 5.105, the zone located at beyond developing a turbulent boundary layer.

  • For a profile of wing, distribution thickness along the Cord (and the gradient of negative pressure associated) of certain profiles known as " laminaires" stabilize the laminarity and allows to move back well the point of transition beyond 5.105: values of 7.106 are possible under nonturbulent aerological conditions (difficult to obtain out of blower) on a perfectly smooth surface (wings of sailplanes).

  • a profiled body as a fuselage (Piaggio P180 Avanti) can have a transition moved back until 50.106, under ideal conditions also.

In medicine

The modifications of mode of flow pulled by the compression of an artery, in general the artery humérale, at the time of the catch of the blood Pressure are responsible for a noise (“noises of Korotkoff”) and allow, by the sounding of the artery downstream from compression, to know the systolic pressure - appearance of the noise, and the diastolic pressure - disappearance of the noise.

The similarity of the fluids

Two flows with equivalent geometry for which Reynolds numbers are equal are known as similar . So that an experiment of small-scale model of a flow gives well a flow similar (i.e. identical to scalings of time, distance and mass near) to the flow in life size, it is necessary that:

Re^ {\ star} = Re \; , \ quad \ quad {p^ {\ star} \ over \ rho^ {\ star} {v^ {\ star}} ^ {2}} = {p \ over \ rho v^ {2}} \; .

The marked values of an asterisk “*” refer to the flow in the small-scale model and the other values with the flow in life size. This is useful for the experiments on the models reduced in liquid vein or wind tunnel where one recovers the data for the flows in real size. For the compressible fluids, the Mach numbers must also be equal for the two fluids so that they can be regarded as equivalents. In a general way, it is necessary that the numbers without dimension characteristic of the flow are identical in the two flows.

See too

Be-X-old: ЛікРэйнальдса

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