In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. 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, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds who popularized its use in 1883 (an example of Stigler's law of eponymy).
Definition The Reynolds number is the ratio of inertial forces to viscous forces within a fluid that is subjected to relative internal movement due to different fluid velocities. A region where these forces change behavior is known as a boundary layer, such as the bounding surface in the interior of a pipe. A similar effect is created by the introduction of a stream of high-velocity fluid into a low-velocity fluid, such as the hot gases emitted from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which tends to inhibit turbulence. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions and is a guide to when turbulent flow will occur in a particular situation.
This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full-size version. Such scaling is not linear and the application of Reynolds numbers to both situations allows scaling factors to be developed.
With respect to laminar and turbulent flow regimes: 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:where: is the density of the fluid (SI units: kg/m3) is the flow velocity (m/s) is a characteristic length (m) is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/(m·s)) is the kinematic viscosity of the fluid (m2/s).
These are generally not independent numbers: liquids generally get less viscous as they become hotter, while gases generally become more viscous. Both will generally become less dense as they become hotter, although the effect is substantially more pronounced in gases. For a (solid) object in a fluid, the characteristic length will vary with temperature - solids are typically even less temperature sensitive than liquids when it comes to density, but at sufficient flow speeds, the amount of frictional heat from traversing the fluid can be substantial.
The Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface.The definition of the Reynolds number is not to be confused with the Reynolds equation or lubrication equation. These definitions generally include the fluid properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension (L in the above equation). 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 for 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.
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 of bounding surfaces can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used.
Derivation If we know that the relevant physical quantities in a physical system are only , then the Reynolds number is essentially fixed by the Buckingham π theorem.
In detail, since there are 4 quantities , but they have only 3 dimensions (length, time, mass), we can consider , where are real numbers. Setting the three dimensions of to zero, we obtain 3 independent linear constraints, so the solution space has 1 dimension, and it is spanned by the vector .
Thus, any dimensionless quantity constructed out of is a function of , the Reynolds number.
Alternatively, we can take the incompressible Navier–Stokes equations (convective form):Remove the gravity term , then the left side consists of inertial force , and viscous force . Their ratio has the order of , the Reynolds number. This argument is written out in detail on the Scallop theorem page.
Alternative derivation The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier–Stokes equations for a newtonian fluid expressed in terms of the Lagrangian derivative:
Each term in the above equation has the units of a "body force" (force per unit volume) with the same dimensions of a density multiplied by an acceleration. Each term is thus dependent on the exact measurements of a flow. When we render the equation nondimensional, that is when we multiply it by a factor with inverse units of the base equation, we obtain a form that does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the factor
where is the mean velocity, or , relative to the fluid (m/s), is the characteristic length (m), is the fluid density (kg/m3).
If we now set
we can rewrite the Navier–Stokes equation without dimensions:
where the term .
Finally, dropping the primes for ease of reading:
This is why mathematically all Newtonian, incompressible flows with the same Reynolds number are comparable. Notice also that in the above equation, the viscous terms vanish for . Thus flows with high Reynolds numbers are approximately inviscid in the free stream.
History
Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes transitioned from laminar flow to turbulent flow.
In his 1883 paper, Reynolds described the transition from laminar to turbulent flow in a classic experiment in which he examined the behaviour of water flow under different flow velocities using a small stream of dyed water introduced into the centre of clear water flow in a larger pipe.
The larger pipe was made of glass so the behaviour of the layer of the dyed stream could be observed. At the end of this pipe, there was a flow control valve used to vary the water velocity inside the tube. When the velocity was low, the dyed layer remained distinct throughout the entire length of the large tube. When the velocity was increased, the layer broke up at a given point and diffused throughout the fluid's cross-section. The point at which this happened was the transition point from laminar to turbulent flow.
From these experiments came the dimensionless Reynolds number for dynamic similarity the ratio of inertial forces to viscous forces. Reynolds also proposed what is now known as the Reynolds averaging of turbulent flows, where quantities such as velocity are expressed as the sum of mean and fluctuating components. Such averaging allows for 'bulk' description of turbulent flow, for example using the Reynolds-averaged Navier–Stokes equations.
Flow in a pipe For flow in a pipe or tube, the Reynolds number is generally defined as
where is the hydraulic diameter of the pipe (the inside diameter if the pipe is circular) (m), is the volumetric flow rate (m3/s), is the pipe's cross-sectional area () (m2), is the mean velocity of the fluid (m/s), (mu) is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s)), (nu) is the kinematic viscosity () (m2/s), (rho) is the density of the fluid (kg/m3), is the mass flowrate of the fluid (kg/s).
For shapes such as squares, rectangular or annular ducts where the height and width are comparable, the characteristic dimension for internal-flow situations is taken to be the hydraulic diameter, , defined as
where is the cross-sectional area, and is the wetted perimeter. The wetted perimeter for a channel is the total perimeter of all channel walls that are in contact with the flow. This means that the length of the channel exposed to air is not included in the wetted perimeter.
For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter:
For an annular duct, such as the outer channel in a tube-in-tube heat exchanger, the hydraulic diameter can be shown algebraically to reduce to
where is the inside diameter of the outer pipe, is the outside diameter of the inner pipe.
For calculation involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy, if the aspect ratio AR of the duct cross-section remains in the range < AR < 4.
Laminar–turbulent transition
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 ≈ , where 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.