# Reynolds number

The nature of the flow of a liquid or gas – laminar or turbulent – is determined by a dimensionless number that depends on the flow rate, the viscosity and density of the liquid, and the characteristic length of the flow element.

Osborne Reynolds was, in a sense, the last adherent of the good old traditions of classical Newtonian mechanics. At the end of his life, he even developed an elaborate mechanical model of the luminiferous ether (cm. Michelson-Morley experiment), according to which the ether was a system of the smallest spherical particles, freely rolling relative to each other like pellets in a bag. Until the end of his days, he believed that “there is no end to the progress of mechanics … and what contemporaries consider it to be a limit and a dead end … in time will only turn out to be a new turn on the path of its development.”

To understand the importance of the main discovery of his life, you first need to talk a little about the so-called dimensionless quantities. Suppose we need to measure the geometric dimensions of a room. Let’s say we took a tape measure and determined that the length of the room is 5 meters. However, if we take a tape measure that is graduated in feet, it turns out that the length of the room is a little over 15 feet. That is, the numbers obtained by us when measuring will depend on the units used, while the real length of the room remains constant.

However, there are some characteristics of the room geometry that do not depend on the units of measurement. In particular, such a value is the ratio of the length of the room to its width – the so-called characteristic ratio… If the room is 20 feet long and 10 feet wide, the aspect ratio is 2. By measuring the length and width of the room in meters, we find that the room is 6.096 m × 3.048 m, but the aspect ratio remains the same: 6.096 m: 3.048 m = 2. In this case, 2 is the dimensionless characteristic of the room.

Now let’s turn to fluid flow. Different fluids, when flowing in pipes, spreading over a surface or flowing around obstacles, have different properties. A thick, sticky liquid (for example, honey) has, as physicists say, more viscosityrather than a light and mobile liquid (such as gasoline). The degree of viscosity of a liquid is determined by the so-called viscosity coefficient, which is usually denoted by the Greek letter η (“this”). Thick, sticky liquids have a viscosity index η tens and hundreds of times higher than that of light and fluid.

Reynolds was able to find a dimensionless number that describes the nature of the flow of a viscous fluid. The scientist himself received it experimentally, after conducting a grueling series of experiments with various liquids, but it was soon shown that it could be derived theoretically from Newton’s laws of mechanics and the equations of classical hydrodynamics. This number, which is now called the Reynolds number and is denoted Re, characterizes the flow and is equal to:

Re = vLρ/η

Where ρ – the density of the liquid, v Is the flow rate, and L Is the characteristic length of the flow element (in this formula it is important to remember that Re Is one number, not a product R × e).

Now let’s look at the dimension of the components of the Reynolds number:

• dimension of the viscosity coefficient η – Newtons multiply by seconds divided by square meters. meters, or n·from/m2… If you remember that, by definition, n = Kg·m/c2, we’ll get Kg/m·from
• density dimension ρ – kilograms divided by cubic meters, or Kg/m3
• dimension of speed v – meters divided by seconds, or m/from
• dimension of stream element length L – meters, or m

Hence, we find that the dimension of the Reynolds number is:

(m/from) × (m) × (Kg/m3): (Kg/m·from)

or, after simplification,

(Kg/m·from): (Kg/m·from)

So, all units of measurement in the dimension of the Reynolds number are canceled, and it really turns out to be a dimensionless quantity.

Reynolds was able to find out that when the value of this number is 2000-3000, the flow becomes completely turbulent, and when the value Re less than a few hundred – the flow is completely laminar (that is, it does not contain swirls). Between these two values, the flow is intermediate.

You can, of course, consider the Reynolds number a purely experimental result, but it can also be interpreted from the standpoint of Newton’s laws. Fluid in a stream has momentum, or, as theorists sometimes say, “inertial force.” In essence, this means that the moving fluid tends to continue its motion at the same speed. In a viscous liquid, this is hindered by the forces of internal friction between the layers of the liquid, which tend to slow down the flow. The Reynolds number just reflects the relationship between these two forces – inertia and viscosity. High values ​​of the Reynolds number describe a situation where the forces of viscosity are relatively small and are not able to smooth out turbulent eddies of the flow. Small values ​​of the Reynolds number correspond to the situation when the forces of viscosity dampen the turbulence, making the flow laminar.

The Reynolds number is very useful from the point of view of modeling flows in various liquids and gases, since their behavior does not depend on the actual viscosity, density, velocity and linear dimensions of the flow element, but only on their ratio, expressed by the Reynolds number. This makes it possible, for example, to place a scaled down model of an aircraft in a wind tunnel and adjust the flow rate so that the Reynolds number corresponds to the actual situation of a full-scale aircraft in flight. (Today, with the development of powerful computer technology, the need for wind tunnels has disappeared, since air flows can be simulated on a computer. In particular, the Boeing 747 was the first civilian airliner entirely designed using computer simulation. that in the design of racing yachts and high-rise buildings, they are still practiced “running” in wind tunnels.) Osborne Reynolds
Osborne Reynolds, 1842-1912

Irish engineer-physicist. Born in Belfast, the son of a hereditary priest of the Church of England. After a short practical training in engineering at a construction company, he entered Cambridge, after which, despite his relative youth, he immediately received the position of professor of civil engineering at Owens College (modern University of Manchester), which he held for 37 years. Reynolds was engaged in scientific and technical developments in the field of hydrodynamics and hydraulics, became the founder of the theories of lubrication and turbulence, and fundamentally improved the design of centrifugal pumps. To study estuarine flows, he built a reduced model of the Mersey River delta.