Reynolds number
In fluid dynamics, the Reynolds number (Re) is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces.[2] At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (eddy currents). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.
The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behavior on a larger scale, such as in local or global air or water movement, and thereby the associated meteorological and climatological effects.
The concept was introduced by George Stokes in 1851,[3] but the Reynolds number was named by Arnold Sommerfeld in 1908[4] after Osborne Reynolds (1842–1912), who popularized its use in 1883.[5][6]
Flow in a wide duct[edit]
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 equal to the distance between the plates.[16] This is consistent with the annular duct and rectangular duct cases above, taken to a limiting aspect ratio.
In order for two flows to be similar, they must have the same geometry and equal Reynolds and Euler numbers. When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds:
where is the Reynolds number for the model, and is full-scale Reynolds number, and similarly for the Euler numbers.
The model numbers and design numbers should be in the same proportion, hence
This allows engineers to perform experiments with reduced scale 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 judgment of the engineer.
An example where the mere Reynolds number is not sufficient for the similarity of flows (or even the flow regime – laminar or turbulent) are bounded flows, i.e. flows that are restricted by walls or other boundaries. A classical example of this is the Taylor–Couette flow, where the dimensionless ratio of radii of bounding cylinders is also important, and many technical applications where these distinctions play an important role.[29][30] Principles of these restrictions were developed by Maurice Marie Alfred Couette and Geoffrey Ingram Taylor and developed further by Floris Takens and David Ruelle.
