Welcome back. In today’s post, I am excited to discuss the Reynolds number, which is one of the most important dimensionless quantities in all of fluid mechanics. By definition, the Reynolds number, denoted *Re*, is the ratio of inertial forces to viscous forces in a fluid, but its significance extends beyond this definition. It is a fundamental quantity that indicates how messy a flow will be. If *Re* is extremely low and close to zero, then a fluid is orderly, and governed by the viscous Stokes equations. The fundamental operator governing the velocity field **u** in these equations is the Laplacian, ∇^{2}, whose solutions satisfy very nice smoothness conditions. Therefore, these extremely low Reynolds number flows cannot become turbulent. If *Re* is large and finite, then a flow is dominated by inertial effects, captured by the nonlinear term **u** • ∇**u**, and has a tendency to become turbulent, through both linear transient energy growth and more complicated nonlinear instabilities. Flows with infinite Reynolds numbers are interesting too. These *inviscid* flows, governed by Euler’s equations, cannot really become turbulent, but they can exhibit some beautiful properties. In particular, the circulation theory that explains how airplane wings create lift is (almost) entirely contained within Euler’s equations. If an inviscid, incompressible flow is also assumed to be irrotational (that is, ∇ × **u** = **0**), Euler’s equations also become completely integrable, and give rise to Bernoulli’s equation, which is a powerful rule that can lead to quick estimates of the flow velocity and pressure in wind tunnels or around airplane wings.

Through these brief examples, I hope I have conveyed that the Reynolds number is a very important quantity that determines where a flow lies on the spectrum from organized Stokes flow (*Re* << 1), to complicated turbulent flow (1 << *Re *< ∞), to slightly less complicated and slightly more beautiful inviscid flow (*Re* = ∞). With this motivation established, we will briefly recall Osborne Reynolds’ 1883 experiment that led to his discovery of the Reynolds number, and then we will present a formal derivation of the Reynolds number that shows how it appears naturally in the Navier-Stokes equations. Finally, we will conclude with a brief preview of subsequent developments in fluid mechanics that leveraged the Reynolds number.

In 1883, Osborne Reynolds (1842-1912) set up an experiment that examined the behavior of pipe flow. In his laboratory, water from a large tank flowed through a smooth, horizontal, clear glass pipe. Colored dye from a reservoir was then injected into the pipe and illuminated the flow as it moved (Reynolds 84-99). When dye was injected in a low-speed flow, the dye would remain smooth, straight, and orderly along a large distance of the pipe downstream from the injection point. However, when dye was injected into a higher-speed flow, the dye would quickly become wavy and diffuse across the pipe only a short distance downstream of the injection point. This disappearance of the dye was caused by the flow becoming turbulent. These observations amounted to a rule that a flow would remain orderly if it traveled slowly or traveled for very short distances, and that a flow would become turbulent if it traveled quickly or traveled over long distances. With a bit more work, Reynolds also recognized that highly viscous fluids (like honey) had a tendency to remain orderly, while fluids with low viscosity (like air) tended to become turbulent. When combined together, these results led to the conclusion that the messiness of a flow was directly proportional to the quantity

*Re* = 𝜌 U L / µ,

where 𝜌 was the fluid density, U was a characteristic flow speed, L was a characteristic length (like the diameter of a pipe), and µ was the dynamic viscosity coefficient of the fluid. While this quantity initially did not have a name, it is now called the Reynolds number, after the person whose work led to its discovery. The pipe flow Reynolds observed generally became turbulent at *Re* > 2300, and remained laminar at *Re* < 2300.

While the significance of the Reynolds number could be empirically observed in Osborne Reynolds’ experiment, it could also be derived from the Navier-Stokes equations that govern all incompressible Newtonian fluids,

𝜌 ∂**u**/∂t + 𝜌**u** • ∇**u** = -∇p + µ∇^{2}**u**

∇ • **u** = 0.

This rigorous justification of the Reynolds number was significant because it showed that the mathematics used to describe fluid flows was consistent with actual behavior measured in the laboratory.

A formal derivation of the Reynolds number from the Navier-Stokes equations proceeded by nondimensionalizing the equations. The motivation for performing this nondimensionalization was that nature did not have a preferred unit system. Therefore, the fundamental physics behind flows were expected to be captured by a nondimensionalized form of the governing equations. If we want to repeat this procedure right now, all dimensional variables can be denoted without primes, while all dimensionless variables can be denoted with primes. Specifically, the physical velocity can be expressed as **u** = U **u’**, where U is a velocity scale. The gradient can be described in terms of a dimensionless gradient as ∇ = 1/L ∇’, where L is a length scale. The pressure can be described in terms of a dimensionless pressure as p = 𝜌U^{2}_{ }p’. Lastly, the time derivative can be nondimensionalized according to the relation ∂/∂t = U/L ∂/∂t’, where L/U is a characteristic time scale. After substituting these relations into the Navier-Stokes equations, the result is

𝜌U^{2}/L ∂**u**’/∂t’ + 𝜌U^{2}/L **u**’ • ∇’**u**’ = – 𝜌U^{2}/L ∇’p’ + µ U/L^{2} ∇’^{2}**u**’

U/L ∇’ • **u**’ = 0.

Dividing the first equation by 𝜌U^{2}/L gives

∂**u**’/∂t’ + **u**’ • ∇’**u**’ = -∇’p’ + µ /(𝜌UL) ∇’^{2}**u**’,

where µ /(𝜌UL) is just the inverse of the Reynolds number. That is to say, the nondimensional Navier-Stokes equations,

∂**u**’/∂t’ + **u**’ • ∇’**u**’ = -∇’p’ + 1/*Re* ∇’^{2}**u**’

∇’ • **u**’ = 0,

are parametrized by a single number, and that number is *Re*. Upon inspecting the first equation above, we can see that *Re* captures the relative significance of the viscous term, which is in-line with our original definition of the Reynolds number as being a ratio of inertial forces to viscous forces. When *Re* is low, 1/*Re* is high, and the Laplacian ∇’^{2} dominates the equations, giving rise to well-behaved dynamics. When *Re* is high, 1/*Re* is low, and the other terms dominate, which give rise to more complicated dynamics because of the nonlinear inertial term **u**’ • ∇’**u**’.

After the Reynolds number was discovered, experimental and theoretical fluid dynamicists used *Re* to characterize the flow regimes they studied. While the applications were too plentiful to give them justice here, the Reynolds number played a prominent role in Ludwig Prandtl’s boundary layer theory of 1904 and the theory of hydrodynamic stability that grew throughout the early 20th century. These theories together revolutionized aerodynamics and inspired research projects that remain active to this day. In future blog posts, we will dive deeper into both of these topics, and learn more about the discoveries the Reynolds number facilitated. Until then, please take care.

Works Cited

Reynolds, Osborne. “An Experimental Investigation of the Circumstances which Determine

whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels.” *Proceedings of the Royal Society of London*, vol. 35, 1883, pp. 84–99. *JSTOR*, http://www.jstor.org/stable/114354. Accessed 2 Sept. 2024.

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