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It is one of the most important Dimensionless Number s in Fluid Dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining Dynamic Similitude . When two geometrically similar flow patterns, in perhaps different fluids with possibly different flowrates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar. It is named after Osborne Reynolds ( 1842 – 1912 ), who proposed it in 1883 .Reynolds, O. 1883. 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. Philosophical Transactions of the Royal Society. 174 pages 935-982 Typically it is given as follows: : where:
For flow in a pipe for instance, the characteristic length is the pipe diameter, if the cross section is circular, or the Hydraulic Diameter , for a non-circular cross section. Laminar Flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while Turbulent Flow , on the other hand, occurs at high Reynolds numbers and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations. The transition between laminar and turbulent flow is often indicated by a critical Reynolds number (''Recrit''), which depends on the exact flow configuration and must be determined experimentally. Within a certain range around this point there is a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and predictions of fluid behaviour can be difficult. For example, within circular pipes the critical Reynolds number is generally accepted to be 2300, where the Reynolds number is based on the pipe diameter and the mean velocity ''v''s within the pipe, but engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 3000 to ensure that the flow is either laminar or turbulent. For flow over a flat plate, the characteristic length is the length of the plate and the characteristic velocity is the free stream velocity. In a boundary layer over a flat plate the local regime of the flow is determined by the Reynolds number based on the distance measured from the leading edge of the plate. In this case, the transition to turbulent flow occurs at a Reynolds number of the order of or . THE SIMILARITY OF FLOWS In order for two flows to be similar they must have the same geometry, and have equal Reynolds numbers and Euler Numbers . When comparing fluid behaviour at homologous points in a model and a full-scale flow, the following holds: : :
L the characteristic length can be best calculated by finding the squares of frontal length and width and then square rooting the sum. REYNOLDS NUMBER SETS THE SMALLEST SCALES OF TURBULENT MOTION In a turbulent flow, there is a range of scales of the fluid motions, sometimes called eddies. A single packet of fluid moving with a bulk velocity is called an Eddy . The size of the largest scales (eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number. As Reynolds number increases, smaller and smaller scales of the flow are visible. In the smoke stack, the smoke may appear to have many very small bumps or eddies, in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in the flow. The higher the Reynolds number, the greater the range of scales. What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important at such scales of the flow. With a strong predominance of inertial forces over viscous forces, the smallest scales of fluid motion are undamped -- there is not enough viscosity to dissipate their motions. The kinetic energy must "cascade" from these large scales to progressively smaller scales until a level is reached for which the scale is small enough for viscosity to become important (that is, viscous forces become of the order of inertial ones). It is at these small scales where the dissipation of energy by viscous action finally takes place. Therefore, although the energy dissipation is produced by a viscous mechanism, the rate at which it occurs is dictated only by large-scale characteristics of the flow, while viscosity only determines the size of the smallest eddies at which the energy will be dissipated. EXAMPLE OF THE IMPORTANCE OF THE REYNOLDS NUMBER If an airplane wing needs testing, one can make a scaled down model of the wing and test it in a wind tunnel using the same Reynolds number that the actual airplane is subjected to. If for example the scale model has linear dimensions one quarter of full size, the flow velocity would have to be ''increased'' four times to obtain similar flow behaviour. Alternatively, tests could be conducted in a water tank instead of in air. As the kinematic viscosity of water is around 13 times less than that of air at 15°C, in this case the scale model would need to be about 13 times smaller in all dimensions to maintain the same Reynolds number, assuming the full-scale flow velocity was used. The results of the laboratory model will be similar to those of the actual plane wing results. Thus there is no need to bring a full scale plane into the lab and actually test it. This is an example of "dynamic similarity". Reynolds number is important in the calculation of a body's Drag characteristics. A notable example is that of the flow around a cylinder. Above roughly 3×106 Re the Drag Coefficient drops considerably. This is important when calculating the optimal cruise speeds for low drag (and therefore long range) profiles for airplanes. REYNOLDS NUMBER IN PHYSIOLOGY Poiseuille's Law on blood circulation in the body are dependent on Laminar Flow . In turbulent flow the flow rate is proportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressure gradient in laminar flow. Using the Reynold's equation we can see that a large diameter, with rapid flow, where the density of the blood is high tends towards turbulence. Rapid changes in vessel diameter may lead to turbulent flow, for instance when a narrower vessel widens to a larger one. Furthermore, an Atheroma may be the cause of turbulent flow, and as such detecting turbulence with a stethoscope may be an indication of such a condition. TYPICAL VALUES OF REYNOLDS NUMBER
Onset of turbulent flow ~ 2.3×103 for pipe flow to 106 for boundary layers
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