Reynolds number
The Reynolds number or Reynolds number (symbol: ) is a named after the physicist Osborne Reynolds dimensionless number. It is used in fluid dynamics and represents the ratio of is (or the ratio of specific Impulskonvektion to pulse diffusion in the system) inertial to viscous forces. It follows that the turbulence behavior similar geometric body is the same for the same Reynolds number. This property allows, for example, realistic model tests in a wind tunnel or water channel.
With
The individual symbols stand for the following sizes:
- - Density characteristic of the fluid ( kg m -3)
- - Flow rate characteristic of the fluid relative to the body (ms -1)
- - Characteristic length of the object ( m)
The characteristic length, also called reference length, can be chosen freely in principle. However, when comparing two currents, this length must be the same type. For flow bodies, the length of the body is chosen in the direction of flow as the reference length usually. In resistor bodies, the width or the height transverse to the usual direction of flow. In pipe flows radius or diameter of the pipe, flumes at the depth or width of the flume surface.
- - Typical dynamic viscosity of the fluid (kg s-1 m-1)
- - Typical kinematic viscosity of the fluid (m2 s-1)
Exceeds the Reynolds number a ( problem dependent ) critical value (), a hitherto laminar flow is sensitive to the smallest disturbances. Accordingly, for with an envelope, the so- called transition to expect from laminar to turbulent flow. Ideal liquids, there is no viscosity and there are no turbulence, and therefore the Reynolds number is infinite.
In magnetohydrodynamics a Reynolds number is also defined: the magnetic Reynolds number.
Applications
The diagram to the right compares speeds and associated Reynolds numbers of some flying objects. For example, the Reynolds number of air vessels are higher than those of aircraft. While you are moving at a slower speed, but are much larger.
The Reynolds number is an important parameter in the similarity theory. If you want to examine, for example, a scale model of an airplane in a wind tunnel, the value of the Reynolds number between original and model must be the same in order to obtain a similar flow field. Accordingly, the ratio must be increased by a factor in a reduced by a factor model. Since the maximum speed is limited, one lowers in cryogenic wind tunnels, in addition, the viscosity of the air by cooling, thereby increasing the density of air at the same time. In this way, Reynolds numbers up to 5 x 107 reached in sample chambers of two meters in diameter. This procedure is very expensive, since usually must be cooled with liquid nitrogen, together with the model of the channel. On cooling, it must be ensured that no form icing. A further increase in the Reynolds number can also be achieved by increasing the static pressure.
Dust grains are very small. If they fall through the air, they have a similar low Reynolds number as a steel ball that falls into a jar of honey. She moves laminar ( ie without vortex formation ) through the fluid. Microorganisms floating at Reynolds numbers 10-5 to 10-2, so that inertial forces are negligible. An example: Heard the flagella of the bacterium E. coli to beat, this would float in less than one atomic diameter to a halt.
In the design of wind turbines, the Reynolds number also plays a role. Through it can stall at the wings determine and thus interpret the system for desired wind speeds.
Examples
Pipe flow
In pipe flows are used as characteristic quantities usually the inner diameter, the amount of the average velocity over the cross section, and the viscosity of the fluid.
We then have: .
In the literature, a value of R = 2300 is often cited. It goes back to measurements of Julius Rotta ( 1950 ).
The critical Reynolds number is not exactly characterized by the transition from laminar to turbulent flow. Rather, decay turbulence below the critical Reynolds number, ie, the faster the smaller the Reynolds number. Succeeded in experiments laminar pipe flow with Reynolds numbers of 50,000 to produce without the flow has become turbulent. If faults produce the envelope into a turbulent flow, the flow is turbulent at supercritical Reynolds number.
The critical Reynolds number, which marks the transition between turbulent and laminar flow is not only dependent upon the geometry of the case of application, but also by the choice of the characteristic length. Is selected as the pipe radius instead of the diameter of the flow as a characteristic linear dimension of a pipe flow, the numerical value to testify the same halved. Since the critical Reynolds number is a value that does not mark a lightning-like envelope, but a broad transition region of the flow conditions, the numerical value is not commonly used but is rounded to.
Channel flow
In channel flow can be used as characteristic quantities of the hydraulic diameter, the magnitude of the average flow velocity over the traversed cross- section, and the viscosity of the fluid.
Rührerströmung
In a stirrer or stirred vessels ( tanks ), the Reynolds number of the linear dimension ( diameter) of the stirrer, the speed or frequency of rotation and on the density and the dynamic viscosity of the liquid is determined:
To avoid confusion, this Reynolds number should be in here with the index for " stirrer ". When the flow at the stirrer is considered turbulent.
Assessment of a turbulent flow
To characterize the degree of turbulence, the Reynolds number can be formed with turbulence- related variables (turbulent Reynolds number ). As characteristic variables, the variance of the velocity and the integral length dimension of the flow then be used, for example. Add to that the (molecular ) viscosity of the fluid.
We then have: .